4 1 solving linear systems by graphing
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This document discusses solving linear systems of equations by graphing. A linear system is a set of two or more linear equations where a solution is an ordered pair that satisfies both equations. Graphing the equations on the same grid allows you to find the point of intersection, which is the solution. Systems can have one solution, infinitely many solutions if the lines are the same, or no solution if the lines are parallel.
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Systems Of Equations
This document provides an overview of solving systems of linear equations by graphing. It begins with examples of identifying whether an ordered pair is a solution to a given system. It then explains that the point of intersection between the graphs of two linear equations is the solution to the corresponding system. The document provides examples of solving systems by graphing and checking solutions by substitution. It concludes with a quiz reviewing the concepts taught.
A1, 6 1, solving systems by graphing (blog 1)
The document defines key terms related to systems of linear equations, including the definition of a system of linear equations as a set of two or more linear equations containing two or more variables. It explains that the solution of a system is an ordered pair that satisfies each equation in the system. Several methods for solving systems are described, including substitution and graphing. Graphing involves setting each equation equal to y and finding the point of intersection of the two lines. Examples are provided to illustrate identifying solutions and solving systems by graphing and checking solutions by substitution.
A1, 6 1, solving systems by graphing (rev)
The document defines key terms related to systems of linear equations, including system notation, solution of a system, and methods for determining if an ordered pair is a solution. It provides examples of identifying solutions by substituting values into systems of equations. Graphing methods for solving systems are described, including rewriting equations in slope-intercept form, graphing the lines, and checking the intersection point as the solution.
Systems of equations by graphing by graphing sect 6 1
This document provides an overview of solving systems of linear equations by graphing. It begins with examples of identifying whether an ordered pair is a solution to a given system. It then explains that the solution to a system is the point of intersection between the graphs of the two equations. Several examples are worked through of solving systems by graphing and checking solutions by substitution. An example word problem application involves setting up and solving a system to determine the number of nights it will take for two girls reading the same book to reach the same number of pages.
Solve By Elimination
The document provides steps for solving a system of equations using the elimination method with addition and subtraction. The key steps are: 1) Put both equations in standard form, 2) Determine which variable to eliminate by finding terms with the same coefficients, 3) Add or subtract the equations to eliminate the chosen variable, 4) Substitute the solved variable back into one equation to find the other, and 5) Check the solution by substituting the values into both original equations. Examples are provided to demonstrate applying these steps to solve systems of equations algebraically.
Solving systems of linear equations by substitution
1) The substitution method for solving systems of linear equations involves solving one equation for one variable and substituting it into the second equation.
2) An example solves the system 2x + y = 4 and 3x + 2y = 7 by solving the first equation for y and substituting it into the second equation.
3) A system can have one solution, infinitely many solutions, or no solutions depending on whether the lines graph the same point, same line, or parallel lines.
Solving Systems by Graphing and Substitution
The document discusses solving systems of linear equations by graphing and substitution. It provides examples of solving systems by graphing, including putting equations in slope-intercept form and finding the intersection point. It also gives examples of solving systems by substitution, including steps like isolating a variable, substituting one equation into the other, solving the resulting equation, and checking solutions. Special cases like no solution and infinitely many solutions are also addressed.
Solving systems of Equations by Elimination
The document provides steps for solving systems of equations using the elimination method with addition and subtraction:
1) Put both equations in standard form.
2) Determine which variable to eliminate by finding terms with the same coefficients.
3) Add or subtract the equations to eliminate the chosen variable.
4) Substitute values back into one equation to find the other variable.
5) Check that the solution satisfies both original equations.
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Systems Of Equations
This document provides an overview of solving systems of linear equations by graphing. It begins with examples of identifying whether an ordered pair is a solution to a given system. It then explains that the point of intersection between the graphs of two linear equations is the solution to the corresponding system. The document provides examples of solving systems by graphing and checking solutions by substitution. It concludes with a quiz reviewing the concepts taught.
A1, 6 1, solving systems by graphing (blog 1)
The document defines key terms related to systems of linear equations, including the definition of a system of linear equations as a set of two or more linear equations containing two or more variables. It explains that the solution of a system is an ordered pair that satisfies each equation in the system. Several methods for solving systems are described, including substitution and graphing. Graphing involves setting each equation equal to y and finding the point of intersection of the two lines. Examples are provided to illustrate identifying solutions and solving systems by graphing and checking solutions by substitution.
A1, 6 1, solving systems by graphing (rev)
The document defines key terms related to systems of linear equations, including system notation, solution of a system, and methods for determining if an ordered pair is a solution. It provides examples of identifying solutions by substituting values into systems of equations. Graphing methods for solving systems are described, including rewriting equations in slope-intercept form, graphing the lines, and checking the intersection point as the solution.
Systems of equations by graphing by graphing sect 6 1
This document provides an overview of solving systems of linear equations by graphing. It begins with examples of identifying whether an ordered pair is a solution to a given system. It then explains that the solution to a system is the point of intersection between the graphs of the two equations. Several examples are worked through of solving systems by graphing and checking solutions by substitution. An example word problem application involves setting up and solving a system to determine the number of nights it will take for two girls reading the same book to reach the same number of pages.
Solve By Elimination
The document provides steps for solving a system of equations using the elimination method with addition and subtraction. The key steps are: 1) Put both equations in standard form, 2) Determine which variable to eliminate by finding terms with the same coefficients, 3) Add or subtract the equations to eliminate the chosen variable, 4) Substitute the solved variable back into one equation to find the other, and 5) Check the solution by substituting the values into both original equations. Examples are provided to demonstrate applying these steps to solve systems of equations algebraically.
Solving systems of linear equations by substitution
1) The substitution method for solving systems of linear equations involves solving one equation for one variable and substituting it into the second equation.
2) An example solves the system 2x + y = 4 and 3x + 2y = 7 by solving the first equation for y and substituting it into the second equation.
3) A system can have one solution, infinitely many solutions, or no solutions depending on whether the lines graph the same point, same line, or parallel lines.
Solving Systems by Graphing and Substitution
The document discusses solving systems of linear equations by graphing and substitution. It provides examples of solving systems by graphing, including putting equations in slope-intercept form and finding the intersection point. It also gives examples of solving systems by substitution, including steps like isolating a variable, substituting one equation into the other, solving the resulting equation, and checking solutions. Special cases like no solution and infinitely many solutions are also addressed.
Solving systems of Equations by Elimination
The document provides steps for solving systems of equations using the elimination method with addition and subtraction:
1) Put both equations in standard form.
2) Determine which variable to eliminate by finding terms with the same coefficients.
3) Add or subtract the equations to eliminate the chosen variable.
4) Substitute values back into one equation to find the other variable.
5) Check that the solution satisfies both original equations.
Solving Systems of Linear Equation using Substitution method
This document describes the substitution method for solving systems of linear equations in two variables. It involves solving one equation for one of the variables and substituting that expression into the other equation to find the value of the other variable. The example shows solving the system of equations x + 3y = -4 and x - y = 16 by first solving the first equation for x in terms of y, then substituting that expression into the second equation to solve for y, and finally substituting the value of y back into the original expression for x to find the solution set (-11, -5).
Systems of Linear Algebra
This document discusses systems of linear equations. It defines a system of linear equations as a set of two or more linear equations. It describes how systems can be analyzed graphically by plotting the lines defined by each equation on a graph and finding their point of intersection. This is the solution to the system. The document also covers the elimination method for solving systems algebraically by adding or subtracting equations to eliminate variables until one is left that can be solved directly. Finally, it introduces the Gaussian elimination method, which uses row operations to systematically transform the coefficients in a system into row echelon form to read the solution directly.
Solving Systems by Substitution
Solving Systems of Equations using Substitution
Step 1) Solve one equation for one variable.
Step 2) Substitute the expression from Step 1 into the other equation.
Step 3) Solve the resulting equation to find the value of the variable. Step 4) Plug this value back into either original equation to find the value of the other variable. Step 5) Check that the solution satisfies both original equations.
Solving Systems - Elimination NOTES
The document discusses the elimination method for solving systems of equations. It explains that the elimination method involves choosing a variable to eliminate, adding or subtracting the equations to cancel out the variable, solving the resulting equation for the remaining variable, and then back-substituting to solve for the other variable. Examples are provided demonstrating how to use both addition and multiplication when eliminating variables to solve systems of two equations.
Solving system of Equations by Graphing
The student will learn to solve systems of equations by graphing. There are three types of solutions when graphing two equations: intersecting lines, which have a single solution where the lines cross; parallel lines, which have no solution; and coinciding lines, which have infinitely many solutions since the lines overlap completely. The student will practice graphing systems of equations and identifying the solution based on whether the lines intersect, are parallel, or coincide.
3 2 solving systems of equations (elimination method)
The document describes the elimination method for solving systems of equations. The key steps are:
1) Write both equations in standard form Ax + By = C
2) Determine which variable to eliminate using addition or subtraction
3) Solve the resulting equation for one variable
4) Substitute back into the original equation to solve for the other variable
5) Check that the solution satisfies both original equations
It provides examples showing how to set up and solve systems of equations using elimination, including word problems about supplementary angles and finding two numbers based on their sum and difference.
8 - solving systems of linear equations by adding or subtracting
Here are the steps to solve this system of equations using elimination by multiplication:
1. Put the equations in standard form:
3x + y = 4
4x + 4y = 6
2. Determine which variable to eliminate. The y-terms have coefficients that are not the same, so multiply the top equation by 4 to make the y-coefficients the same:
12x + 4y = 16
4x + 4y = 6
3. Subtract the equations to eliminate y:
12x + 4y = 16
- (4x + 4y = 6)
8x = 10
x = 5/4
4. Plug x back into an original equation and
Solving Systems of Linear Equations by Graphing
Here are the solutions to the systems of equations without graphing:
1. One solution (different slopes)
2. No solution (same slope, different intercepts)
3. One solution (different slopes)
4. No solution (same slope, different intercepts)
5. One solution (different slopes)
6. Infinitely many solutions (same slope and intercept)
7. One solution (different slopes)
8. No solution (same slope, different intercepts)
9. One solution (different slopes)
Solve Systems By Elimination
1) Put the equations in standard form if needed
2) Determine that y can be eliminated by multiplying the top equation by 3
3) Multiply the top equation by 3 and add it to the bottom equation to eliminate y
4) Solve the resulting equation for the remaining variable
5) Substitute back into one of the original equations to find the other variable
The solution to the system is (1, -2).
Elimination method Ch 7
The document discusses the elimination method for solving systems of equations. It provides examples of using addition or multiplication to eliminate a variable and obtain a solution. Key steps include adding or multiplying the equations appropriately, substituting values back to solve for the remaining variable, and obtaining the solution point (x,y).
Systems of equations
This document covers systems of linear and nonlinear equations. It discusses various methods for solving systems algebraically, including substitution, elimination, and graphing. It defines the three possible solutions to a system of two linear equations: a unique solution, no solution (inconsistent), or infinitely many solutions (dependent). Examples are provided to illustrate each of these cases and solving systems is applied to real-world word problems. Key concepts covered include linear and nonlinear systems, consistent and inconsistent systems, dependent and independent systems, and various algebraic techniques for solving different types of systems.
9.1 Systems of Linear Equations
This document discusses systems of linear equations and methods for solving them. It defines a linear system as a set of equations where all variables have an exponent of 1. There are three possibilities for a system: 1) a single solution, 2) no solution (inconsistent), or 3) infinitely many solutions. Four methods are presented for solving systems: substitution, elimination, graphing, and matrices. Examples are provided to illustrate substitution and elimination. The document also discusses how to determine if a system is inconsistent or has infinitely many solutions based on the outcome of solving the system.
Solving systems of equations by substitution
ejaymathact.wordpress.com
Disclaimer: Some parts of the presentation are obtained from various sources. Credit to the rightful owners.
Systemsof3 equations
This document discusses solving systems of three linear equations in three variables using the elimination method. It provides an example of using elimination to solve the system of equations x - 3y + 6z = 21, 3x + 2y - 5z = -30, and 2x - 5y + 2z = -6. The steps are: 1) rewrite the system as two smaller systems with two equations each, 2) eliminate the same variable from each smaller system, 3) solve the resulting system of two equations for the two remaining variables, 4) substitute back into one of the original equations to find the third variable, and 5) check that the solution satisfies all three original equations. The solution to the example system is (-
15.2 solving systems of equations by substitution
This document provides examples for solving systems of linear equations by substitution. It begins with an example showing the step-by-step process for solving a system using substitution. Subsequent examples demonstrate how to solve special systems that have no solution, infinitely many solutions, and systems with fractions or decimals. The final example walks through solving a system with fractions using clear fractions before substitution.
Solving System of Equations by Substitution
1) The document discusses solving systems of equations using substitution. It provides 5 steps for solving a system by substitution: 1) solve one equation for a variable, 2) substitute into the other equation, 3) solve the new equation, 4) plug back in to find the other variable, and 5) check the solution.
2) It then works through examples, showing that substitution is easiest when one equation is already solved for a variable. It also notes that if the final step results in a false statement, there are no solutions, and if true, there are infinitely many solutions.
System of linear equations and their solution
1. The document presents examples of solving addition and subtraction problems with positive and negative numbers. It also gives examples of finding two numbers whose sum and difference satisfy given values.
2. It defines a system of linear equations as two equations in two variables of the form Ax + By = C and Dx + Ey = F. It explains that an ordered pair (x,y) is a solution if it satisfies both equations.
3. Examples are given of determining if an ordered pair is a solution to a given system of linear equations by substituting the values into each equation and checking if both equations are satisfied. An activity asks students to identify if ordered pairs are solutions to several systems.
Solving linear systems by the substitution method
The document discusses solving systems of linear equations using substitution and elimination methods. It provides 4 examples of solving systems of 2 equations with 2 unknowns. The substitution method involves solving one equation for one variable in terms of the other and substituting it into the second equation. The elimination method involves multiplying equations by constants and adding/subtracting them to eliminate one variable. Both methods are shown to yield the solution point that satisfies both equations.
3.2 Solving Linear Systems Algebraically
This document provides examples of solving systems of linear equations using substitution and combination methods. It demonstrates solving 3x+4y=-4 and x+2y=2 by substitution, solving -x+3y=1 and 4x+6y=8 by substitution, and solving 2x-4y=13 and 4x-5y=8 by combination. It also shows multiplying equations by constants before combining, and solving a 4 equation system.
Systems of 3 Equations in 3 Variables
The document discusses solving systems of 3 linear equations with 3 variables. It provides steps to set up the equations in standard form, eliminate one variable using two equations, eliminate the same variable from another pair of equations to get a system of 2 equations with 2 variables, solve this system to find the values of two variables, substitute these values into the original third equation to solve for the remaining variable, and check the solution. An example problem demonstrates applying these steps to solve for x, y, and z. The summary notes that the solution may not be unique depending on whether eliminating variables results in a true or false statement.
Solving systems of Linear Equations
There are three possible solutions to a system of linear equations in two variables:
One solution: the graphs intersect at a single point, giving the solution coordinates.
No solution: the graphs are parallel lines, making the system inconsistent.
Infinitely many solutions: the graphs are the same line, making the equations dependent.
The substitution method for solving systems involves: 1) solving one equation for a variable, 2) substituting into the other equation, 3) solving the new equation, and 4) back-substituting to find the remaining variable.
Solve systemsbygraphing
This document discusses solving systems of equations by graphing. A system of equations contains two or more equations with the same variables. When graphing systems, lines can intersect at a point, be parallel with no solution, or coincide with infinitely many solutions. The document provides examples of each case and instructs students to graph systems, identify the intersection point or lack thereof, and check their answer by substituting values back into the original equations.
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This document describes the substitution method for solving systems of linear equations in two variables. It involves solving one equation for one of the variables and substituting that expression into the other equation to find the value of the other variable. The example shows solving the system of equations x + 3y = -4 and x - y = 16 by first solving the first equation for x in terms of y, then substituting that expression into the second equation to solve for y, and finally substituting the value of y back into the original expression for x to find the solution set (-11, -5).
Systems of Linear Algebra
This document discusses systems of linear equations. It defines a system of linear equations as a set of two or more linear equations. It describes how systems can be analyzed graphically by plotting the lines defined by each equation on a graph and finding their point of intersection. This is the solution to the system. The document also covers the elimination method for solving systems algebraically by adding or subtracting equations to eliminate variables until one is left that can be solved directly. Finally, it introduces the Gaussian elimination method, which uses row operations to systematically transform the coefficients in a system into row echelon form to read the solution directly.
Solving Systems by Substitution
Solving Systems of Equations using Substitution
Step 1) Solve one equation for one variable.
Step 2) Substitute the expression from Step 1 into the other equation.
Step 3) Solve the resulting equation to find the value of the variable. Step 4) Plug this value back into either original equation to find the value of the other variable. Step 5) Check that the solution satisfies both original equations.
Solving Systems - Elimination NOTES
The document discusses the elimination method for solving systems of equations. It explains that the elimination method involves choosing a variable to eliminate, adding or subtracting the equations to cancel out the variable, solving the resulting equation for the remaining variable, and then back-substituting to solve for the other variable. Examples are provided demonstrating how to use both addition and multiplication when eliminating variables to solve systems of two equations.
Solving system of Equations by Graphing
The student will learn to solve systems of equations by graphing. There are three types of solutions when graphing two equations: intersecting lines, which have a single solution where the lines cross; parallel lines, which have no solution; and coinciding lines, which have infinitely many solutions since the lines overlap completely. The student will practice graphing systems of equations and identifying the solution based on whether the lines intersect, are parallel, or coincide.
3 2 solving systems of equations (elimination method)
The document describes the elimination method for solving systems of equations. The key steps are:
1) Write both equations in standard form Ax + By = C
2) Determine which variable to eliminate using addition or subtraction
3) Solve the resulting equation for one variable
4) Substitute back into the original equation to solve for the other variable
5) Check that the solution satisfies both original equations
It provides examples showing how to set up and solve systems of equations using elimination, including word problems about supplementary angles and finding two numbers based on their sum and difference.
8 - solving systems of linear equations by adding or subtracting
Here are the steps to solve this system of equations using elimination by multiplication:
1. Put the equations in standard form:
3x + y = 4
4x + 4y = 6
2. Determine which variable to eliminate. The y-terms have coefficients that are not the same, so multiply the top equation by 4 to make the y-coefficients the same:
12x + 4y = 16
4x + 4y = 6
3. Subtract the equations to eliminate y:
12x + 4y = 16
- (4x + 4y = 6)
8x = 10
x = 5/4
4. Plug x back into an original equation and
Solving Systems of Linear Equations by Graphing
Here are the solutions to the systems of equations without graphing:
1. One solution (different slopes)
2. No solution (same slope, different intercepts)
3. One solution (different slopes)
4. No solution (same slope, different intercepts)
5. One solution (different slopes)
6. Infinitely many solutions (same slope and intercept)
7. One solution (different slopes)
8. No solution (same slope, different intercepts)
9. One solution (different slopes)
Solve Systems By Elimination
1) Put the equations in standard form if needed
2) Determine that y can be eliminated by multiplying the top equation by 3
3) Multiply the top equation by 3 and add it to the bottom equation to eliminate y
4) Solve the resulting equation for the remaining variable
5) Substitute back into one of the original equations to find the other variable
The solution to the system is (1, -2).
Elimination method Ch 7
The document discusses the elimination method for solving systems of equations. It provides examples of using addition or multiplication to eliminate a variable and obtain a solution. Key steps include adding or multiplying the equations appropriately, substituting values back to solve for the remaining variable, and obtaining the solution point (x,y).
Systems of equations
This document covers systems of linear and nonlinear equations. It discusses various methods for solving systems algebraically, including substitution, elimination, and graphing. It defines the three possible solutions to a system of two linear equations: a unique solution, no solution (inconsistent), or infinitely many solutions (dependent). Examples are provided to illustrate each of these cases and solving systems is applied to real-world word problems. Key concepts covered include linear and nonlinear systems, consistent and inconsistent systems, dependent and independent systems, and various algebraic techniques for solving different types of systems.
9.1 Systems of Linear Equations
This document discusses systems of linear equations and methods for solving them. It defines a linear system as a set of equations where all variables have an exponent of 1. There are three possibilities for a system: 1) a single solution, 2) no solution (inconsistent), or 3) infinitely many solutions. Four methods are presented for solving systems: substitution, elimination, graphing, and matrices. Examples are provided to illustrate substitution and elimination. The document also discusses how to determine if a system is inconsistent or has infinitely many solutions based on the outcome of solving the system.
Solving systems of equations by substitution
ejaymathact.wordpress.com
Disclaimer: Some parts of the presentation are obtained from various sources. Credit to the rightful owners.
Systemsof3 equations
This document discusses solving systems of three linear equations in three variables using the elimination method. It provides an example of using elimination to solve the system of equations x - 3y + 6z = 21, 3x + 2y - 5z = -30, and 2x - 5y + 2z = -6. The steps are: 1) rewrite the system as two smaller systems with two equations each, 2) eliminate the same variable from each smaller system, 3) solve the resulting system of two equations for the two remaining variables, 4) substitute back into one of the original equations to find the third variable, and 5) check that the solution satisfies all three original equations. The solution to the example system is (-
15.2 solving systems of equations by substitution
This document provides examples for solving systems of linear equations by substitution. It begins with an example showing the step-by-step process for solving a system using substitution. Subsequent examples demonstrate how to solve special systems that have no solution, infinitely many solutions, and systems with fractions or decimals. The final example walks through solving a system with fractions using clear fractions before substitution.
Solving System of Equations by Substitution
1) The document discusses solving systems of equations using substitution. It provides 5 steps for solving a system by substitution: 1) solve one equation for a variable, 2) substitute into the other equation, 3) solve the new equation, 4) plug back in to find the other variable, and 5) check the solution.
2) It then works through examples, showing that substitution is easiest when one equation is already solved for a variable. It also notes that if the final step results in a false statement, there are no solutions, and if true, there are infinitely many solutions.
System of linear equations and their solution
1. The document presents examples of solving addition and subtraction problems with positive and negative numbers. It also gives examples of finding two numbers whose sum and difference satisfy given values.
2. It defines a system of linear equations as two equations in two variables of the form Ax + By = C and Dx + Ey = F. It explains that an ordered pair (x,y) is a solution if it satisfies both equations.
3. Examples are given of determining if an ordered pair is a solution to a given system of linear equations by substituting the values into each equation and checking if both equations are satisfied. An activity asks students to identify if ordered pairs are solutions to several systems.
Solving linear systems by the substitution method
The document discusses solving systems of linear equations using substitution and elimination methods. It provides 4 examples of solving systems of 2 equations with 2 unknowns. The substitution method involves solving one equation for one variable in terms of the other and substituting it into the second equation. The elimination method involves multiplying equations by constants and adding/subtracting them to eliminate one variable. Both methods are shown to yield the solution point that satisfies both equations.
3.2 Solving Linear Systems Algebraically
This document provides examples of solving systems of linear equations using substitution and combination methods. It demonstrates solving 3x+4y=-4 and x+2y=2 by substitution, solving -x+3y=1 and 4x+6y=8 by substitution, and solving 2x-4y=13 and 4x-5y=8 by combination. It also shows multiplying equations by constants before combining, and solving a 4 equation system.
Systems of 3 Equations in 3 Variables
The document discusses solving systems of 3 linear equations with 3 variables. It provides steps to set up the equations in standard form, eliminate one variable using two equations, eliminate the same variable from another pair of equations to get a system of 2 equations with 2 variables, solve this system to find the values of two variables, substitute these values into the original third equation to solve for the remaining variable, and check the solution. An example problem demonstrates applying these steps to solve for x, y, and z. The summary notes that the solution may not be unique depending on whether eliminating variables results in a true or false statement.
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Solving systems of Linear Equations
There are three possible solutions to a system of linear equations in two variables:
One solution: the graphs intersect at a single point, giving the solution coordinates.
No solution: the graphs are parallel lines, making the system inconsistent.
Infinitely many solutions: the graphs are the same line, making the equations dependent.
The substitution method for solving systems involves: 1) solving one equation for a variable, 2) substituting into the other equation, 3) solving the new equation, and 4) back-substituting to find the remaining variable.
Solve systemsbygraphing
This document discusses solving systems of equations by graphing. A system of equations contains two or more equations with the same variables. When graphing systems, lines can intersect at a point, be parallel with no solution, or coincide with infinitely many solutions. The document provides examples of each case and instructs students to graph systems, identify the intersection point or lack thereof, and check their answer by substituting values back into the original equations.
Solve systemsbygraphing
The document discusses solving systems of equations by graphing. It defines a system of equations as two or more equations using the same variables. When graphing systems, there can be three possibilities: intersecting lines which have a single solution where the lines cross; parallel lines which have no solution; or coinciding lines which have infinitely many solutions since the lines overlap. The document provides examples of solving systems by graphing and checking solutions by substituting point values back into the original equations.
Solve systemsbygraphing
This document discusses solving systems of equations by graphing. It defines a system of equations as two or more equations using the same variables, where the solution satisfies all equations. There are three possibilities when graphing systems: intersecting lines, where the intersection point is the solution; parallel lines, where there is no solution; and coinciding lines, where there are infinitely many solutions. It provides examples of each type of system and guides the student through solving sample systems by graphing and checking solutions.
3.1 Solving Linear Systems By Graphing
This document discusses systems of linear equations and inequalities, including: defining a system as a collection of equations dealt with together; solutions being ordered pairs that satisfy all equations; examples of determining if points are solutions; solutions being intersections of lines; determining the number of solutions based on whether lines intersect, are the same, or are parallel.
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This document discusses linear systems and methods for solving them. A linear system is a set of linear equations where a solution is an ordered pair that satisfies both equations. Graphically, the solution is the point where the lines intersect. Systems can have one solution, infinitely many solutions if the lines are the same, or no solution if the lines are parallel. The document presents two methods for algebraically solving systems: substitution and elimination. Substitution involves solving one equation for a variable and substituting it into the other equation. Elimination involves adding or subtracting equations to eliminate a variable.
Systems of Linear Equations Graphing
There are three possible outcomes when graphing two linear equations:
1) A single point of intersection, meaning one solution.
2) Parallel lines, meaning no solution.
3) Coincident lines with the same slope and y-intercept, meaning an infinite number of solutions.
The type of solution can be determined without graphing by writing the equations in slope-intercept form and comparing slopes and y-intercepts. Different slopes means one solution, same slope but different intercepts means no solution, and same slope and intercept means infinite solutions.
Intro to systems_of_linear_equations
1) A system of linear equations is two or more linear equations whose solution is being sought. The solution is the ordered pair(s) that satisfy all equations.
2) There are three types of systems: consistent if lines intersect at one point (one solution), inconsistent if lines are parallel (no solution), and dependent if equations graph the same line (infinite solutions).
3) In the examples, (-4,16) is the solution to one consistent system while (-2,3) is not a solution to another since the equations are not equal at that point.
Solve systemsbygraphing
A system of equations involves two or more equations with the same variables. When graphing systems of equations, there are three possibilities for the graphs: intersecting lines which have a single solution, parallel lines which have no solution, and coinciding lines which have infinitely many solutions. To solve a system using graphs, graph both equations and find the point of intersection, which is the solution, then check it satisfies both equations.
January18
The document provides information about solving systems of linear equations through three main methods: graphing, elimination by addition, and elimination by multiplication. It includes examples of using each method to solve systems with steps shown for substitution and verification of solutions. Practice problems are presented for students to determine the number of solutions from graphs of systems and to solve systems using the elimination methods.
Linear equations rev - copy
A system of linear equations in two variables can be solved either graphically or algebraically. Graphically, the solutions are found by drawing the lines corresponding to each equation and finding their point(s) of intersection. Algebraically, the equations are combined to eliminate one variable, resulting in an equation that can be solved for the remaining variable. A system has a single solution if the lines intersect at one point, no solution if the lines are parallel, or infinite solutions if the lines coincide as the same line.
Pair of linear equations
A system of linear equations in two variables can be solved either graphically or algebraically. Graphically, the solutions are found by drawing the lines corresponding to each equation and finding their point(s) of intersection. Algebraically, the equations are combined to eliminate one variable, resulting in an equation that can be solved for the remaining variable. A system has a single solution if the lines intersect at one point, no solution if the lines are parallel, or infinite solutions if the lines coincide as the same line.
January 29, 2014
Let x = the smaller number
y = the larger number
Then: x + y = 52 (the sum is 52)
y = 4x + 2 (the larger is 4 times the smaller plus 2)
Substitute the second equation into the first:
x + (4x + 2) = 52
5x + 2 = 52
5x = 50
x = 10
y = 4(10) + 2 = 42
The numbers are 10 and 42.
Linear equations rev
A system of linear equations in two variables can be solved either graphically or algebraically. Graphically involves drawing the lines and finding their point of intersection. Algebraically involves combining the equations to eliminate variables until one is left. A system has a single solution if the lines intersect, no solution if parallel, and infinite solutions if coincident. Algebraic methods include substitution and elimination to solve for the variables.
February 11, 2015,
This document provides instruction on solving systems of linear equations and inequalities. It introduces systems of equations, discusses three methods for solving them (graphing, elimination, and substitution), and provides examples of each method. It also assigns practice problems for students to work on solving systems by graphing. Finally, it previews content on inequalities that will be covered tomorrow and assigns a test on coordinate planes for the next class.
M1 L5 Remediation Notes
This document discusses solving systems of equations and inequalities through three main methods: graphing, substitution, and elimination. It provides examples of each method. For graphing systems, it explains the three possibilities for the graphs: consistent systems with one solution where the lines intersect, inconsistent systems with no solution where the lines are parallel, and dependent systems with infinite solutions where the lines coincide. It then works through examples of using substitution and elimination to solve systems algebraically. [/SUMMARY]
Solving Systems of Equations-graphing and Elimination
This document provides an overview of solving systems of linear equations through graphing and algebra. It defines a system of equations as a set of two or more equations and describes the different types of solutions as unique (intersecting lines), inconsistent (parallel lines with no solution), or dependent (same line with infinitely many solutions). Students are given examples to solve systems of equations through both graphing and algebraic methods.
Systems equations two varibles
The document discusses solving systems of linear equations graphically. It explains that systems can have one solution, no solution, or infinitely many solutions depending on whether the lines intersect at one point, are parallel, or are the same line. Several examples are worked through showing how to graph the lines and determine the number of solutions. The key steps are: 1) write the equations in slope-intercept form, 2) graph them on the same plane, 3) find where they intersect, and 4) check the solution satisfies both equations.
Linear Algebra - systems of equations (week 1).ppt
The document provides an overview of solving systems of linear equations by graphing and algebraically through substitution and elimination. It includes examples of solving systems by each method and determining the number of solutions based on whether the lines intersect, are parallel, or are identical. Key steps for each algebraic method are outlined. Systems can have one solution, no solution, or an infinite number of solutions depending on the relationship between the lines.
systems of equations.ppt
The document provides an overview of solving systems of linear equations by graphing and algebraically through substitution and elimination. It includes examples of using each method to solve systems. Key points covered include the three possible outcomes when graphing systems, the steps for the substitution and elimination methods, and checking solutions in the original equations.
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Nth roots can be expressed using rational exponents, where the denominator of the exponent is the index of the radical. To evaluate expressions with rational exponents, one should rewrite them in radical form, evaluate the radical, and then raise the expression to the power indicated by the original exponent. Examples are provided of evaluating various expressions with rational exponents.
6.6 quadratic formula
The document introduces the quadratic formula as a method for solving any quadratic equation in the standard form of ax^2 + bx + c = 0. It provides the quadratic formula, and shows an example of using it to solve for x in a quadratic equation. Users are then prompted to try solving for x in a quadratic equation using the quadratic formula.
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This document discusses classifying second-degree equations as different types of conic sections. It provides the general form of a second-degree equation, explains how to use the discriminant to determine if the graph is a circle, ellipse, parabola, or hyperbola, and gives examples of identifying the conic section from different equations. Key aspects covered include degenerate conics, the general form containing terms for x2, xy, y2, x, y, and the constant, and using the discriminant calculated from the equation coefficients to classify the conic section.
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4 1 solving linear systems by graphing
- 1. 4-1 SOLVING LINEAR SYSTEMS BY GRAPHING
- 2. WHAT IS A LINEAR SYSTEM? A set of two or more linear equations. A solution is an ordered pair (x, y) that makes BOTH equations true. Happens graphs at the intersection of the
- 3. CHECKING SOLUTIONS Example: Is 3x x Is (2, 2) a solution of the system? 2y 2y 2 6 (0, -1) a solution?
- 4. SOLVING BY GRAPHING 1. 2. 3. Graph each equation on the same grid Estimate the point of intersection Check solution in original system (BOTH equations!)
- 5. EXAMPLE Solve y = 2x + 5 y = -4x – 1
- 6. YOUR TURN! Solve y=x–1 y = -x + 3
- 7. MANY SOLUTIONS When will a system have infinitely many solutions? When both equations are the same line! Example: y = -5x – 2 5x + y = -2
- 8. NO SOLUTIONS When will a system have no solution? When the lines are parallel! Example: y = 3x + 1 y = 3x – 5
- 9. SUMMARY Lines intersect at one point. One Solution Both are same line. Infinitely Many Solutions Two parallel lines. No Solution