1.8 linear systems
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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.
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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.
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I made this presentation for my own college assignment and i had referred contents from websites and other presentations and made it presentable and reasonable hope you will like it!!!
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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
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x = 5/4
4. Plug x back into an original equation and
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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
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5) Substitute back into one of the original equations to find the other variable
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Here are the steps to solve this system of equations using elimination by multiplication:
1. Put the equations in standard form:
3x + y = 4
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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:
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This document provides an overview of solving systems of linear equations algebraically. There are two main methods: substitution and elimination. Substitution involves isolating one variable and substituting it into the other equation. Elimination involves adding or subtracting equations to eliminate one variable. Examples are provided to demonstrate both methods. The objectives are for students to be able to solve linear systems algebraically and relate it to representing relationships between quantities with graphs and equations.
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7.3 rational exponents
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.
6 7 new look at conics
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.
6 6 systems of second degree equations
This document discusses methods for solving systems of second-degree equations, including algebraic substitution or elimination methods and graphical methods. It provides 3 examples of solving systems of equations algebraically and graphically, including solving systems of quadratic equations and systems involving circles.
4.4 multi step trig problems
This document provides instructions for solving for the length of a shared side (x) of two triangles. It instructs the user to write two equations, one for each triangle, relating the lengths of the sides including x. It also describes an approach for solving when the triangles form one large non-right triangle, which is to draw the altitude splitting it into two triangles and using the fact that triangle angles sum to 180 degrees. The user is then told to solve the two equations to find the value of x.
6 5 parabolas
This document provides an overview of parabolas including their key properties and examples of how to graph and write equations of parabolas. It defines that every point on a parabola is equidistant from the focus and directrix, with the directrix perpendicular to the line of symmetry and the vertex halfway between them. Examples are given of finding the focus, directrix, and graphing parabolas given these features as well as writing the equation of a parabola given its directrix.
4.3 finding missing angles
Inverse trig ratios, including sin-1, cos-1, and tan-1, are used to solve for missing angle measures. When given a decimal value for sin, cos, or tan, the corresponding inverse trig function is used to find the angle measure to the nearest degree. When given a triangle, the appropriate trig ratio is set up and the inverse function is then used to find the measure of the indicated angle to the nearest degree.
4.2 solving for missing sides
This document provides instructions for solving for a missing side of a triangle using trigonometric functions. It tells the reader to label the sides of the triangle, set up an equation using the information given, and solve for the variable to find the missing side, making sure a calculator is in degree mode.
4.1 trig ratios
This document defines and explains the three main trigonometric ratios - sine, cosine, and tangent - which are used to solve for missing angles and sides of right triangles. It identifies the hypotenuse as the longest side, and the adjacent and opposite sides in relation to a given angle. The document provides the ratio equations and examples of calculating trig ratios for various angles in a right triangle.
R.4 solving literal equations
This document defines a literal equation as an equation with two or more different variables. It provides examples of literal equations such as A = bh, where b is solved for, and P = 2L + 2W, where W is solved for. The document demonstrates how to solve various literal equations for the indicated variable, such as solving for P in I = Prt, solving for x in A = x + y^2, and solving for x and y in two additional equations.
R.3 solving 1 var inequalities
This document provides instructions for solving inequalities and graphing their solutions. It notes that when multiplying or dividing an inequality by a negative number, the inequality symbol is reversed. Open circles are used for < and >, while closed circles are used for ≤ and ≥. Examples are provided of solving various inequalities algebraically and graphing the resulting solution sets on a number line.
R.2 solving multi step equations
Multi-step equations involve more than two steps to solve. To solve them, simplify the side with variables by distributing and combining like terms, then solve as a single-step or two-step equation. The key steps are to distribute any parentheses, collect like variable terms on one side and constants on the other, and use opposite operations to solve for the variable.
R.1 simplifying expressions
Algebraic expressions contain numbers, variables, and operators and can be simplified using the distributive property and by combining like terms. The document provides examples of algebraic expressions that can be simplified by first distributing all terms inside parentheses and then combining like terms, such as 3(4n + 2) + n, which is simplified to 12n + 6 + n = 13n + 6.
10 4 solving trig equations
This document discusses two methods for solving trigonometric equations: graphing and algebraically. Graphing allows for visually seeing solutions but may result in decimal answers, while algebraic solving provides exact solutions but can become messy. It provides tips for algebraic solving based on whether the equation contains functions of 2x and x, or only 2x. Examples are given to demonstrate each method.
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7 3 multiplying polynomials
This document discusses four methods for multiplying binomial expressions: algebra tiles, the distributive property, a table or "box" method, and FOIL. It explains that every term in the first binomial must be multiplied by every term in the second binomial. Examples are provided of multiplying binomials such as (x + 2)(x + 5), along with practice problems like (x + 3)(x - 4) and (2p - 4)(3p + 2).
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The document discusses simplifying radical expressions. It defines key terms like the radical sign and radicand. It provides two methods for simplifying radicals: using the product property to rewrite with the largest perfect square factor or making a factor tree to pull out factors. Examples are provided to demonstrate simplifying radicals of various forms, including those with variables and higher root expressions. The document also contains practice problems for simplifying radical expressions.
10 3 double and half-angle formulas
1) This document discusses double-angle and half-angle formulas for trigonometric functions like sine, cosine, and tangent.
2) It derives formulas that relate trig functions of double and half angles to trig functions of the original angle, such as sin(2x) = 2sin(x)cos(x) and sin(x/2) = ±√(1-cos(x))/2.
3) Examples are provided to demonstrate applying these formulas to simplify trigonometric expressions and derive new identities.
6.7 other methods for solving
This document discusses two additional methods for solving quadratic equations: the quadratic formula and graphing. The quadratic formula can be used to solve any quadratic equation in standard form (ax^2 + bx + c = 0) by inputting the a, b, and c coefficients. Graphing involves changing the equation to y= form, plotting it on a calculator to find the x-intercepts, which are the solutions. Both methods are demonstrated with examples.
10 2 sum and diff formulas for tangent
1) Formulas are derived for tangent of the sum and difference of two angles using previous trigonometric identities: tan(α + β) = (tanα + tanβ) / (1 - tanαtanβ) and tan(α - β) = (tanα - tanβ) / (1 + tanαtanβ).
2) These formulas are valid when the tangents of the individual angles and their sum/difference are defined, which excludes cases where the cosine of any angle is 0.
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1.8 linear systems
- 2. What is a Linear System? •A set or collection of linear equations. • A solution of the system is an ordered pair (x, y) that makes BOTH equations true.
- 3. Checking Solutions • Example: • Is (2, 2) a solution of the system? x y 3 2 2 x 2 y 6 • Is (0, -1) a solution?
- 4. Solving Systems Graphically • Solution of one equation: any point on the line. • Solution of a system: point that falls on BOTH lines. • Example: y x 3 2 6 y x
- 5. Example • Graph the system and find the solution. y = x + 4 y = -x + 2
- 6. You Try! • Graph the system and find the solution. y = -4x + 2 y = x – 3
- 7. Many Solutions • When will a system have infinitely many solutions? ▫ When both equations are the same line! Example: x y 3 2 6 x y 6 4 12
- 8. No Solutions • When will a system have no solution? ▫ When the lines are parallel! Example: x y 3 2 6 x y 3 2 2
- 9. Lines intersect at one point. One Solution Both are same line. Infinitely Many Solutions Two parallel lines. No Solution Summary
- 10. Substitution Method Example: 3x + 4y = -4 x + 2y = 2 • 1. Solve one equation for one variable. • 2. Substitute that into the other equation and solve. • 3. Plug in answer and solve for the other variable.
- 11. Example 2 • Solve using substitution. 2x – 3y = -1 y = x – 1
- 12. You Try! • Solve the linear system using substitution. 3x – y = 13 2x + 2y = -10
- 13. Elimination Method Example: -3x + 7y = -16 -9x + 5y = 16 • 1. Look for a variable that will cancel if equations are added. • 2. If none, multiply one or both equations by a constant. • 3. Add equations to eliminate one variable. Solve for the other. • 4. Plug it in to solve for other variable.
- 14. Example: • Solve the system using elimination. 7x + 2y = 24 8x + 2y = 30
- 15. You Try! • Solve the system using elimination. 3x + 2y = 6 -6x - 3y = -6
- 16. Example: • Solve the system. 7x - 12y = -22 -5x + 8y = 14
- 17. You Try! • Solve the system using elimination. 9x – 5y = -7 -6x + 4y = 2