Systems of Equations Presented by: Cara VaronEDUU 551 Applications of Computers Brandman University
Activating Schema• Chapters 2 and 3: You learned to solve multi-step problems involving linear equations and inequalities in one variable and provided justification for each step.• Chapter 5: You learned how to graph a linear equation and compute the x- and y- intercepts• Chapter 5: You learned to verify that a point lies on a line, given the equation of the line.
VocabularyDefinitions1. A system of linear equations is a situation in which two or more linear equations are together in the same scenario2. A solution of the system of linear equations is an ordered pair in a system that makes all of the equations true.
Solving Systems by GraphingSteps to follow:1. Graph both equations on the same coordinate plane2. Find the point of intersection3. Check to see if the point of intersection makes both equations true
Three Possible Solutions of Systems of Linear Equations• If the lines have different slopes, then the lines intersect, so there is only one solution• If the lines have the same slope and different y- intercepts, then the lines are parallel and there are no solutions• If the lines have the same slope and the same y-intercept, then the lines are the same, so there are infinite solutions
Solving Systems Using Substitution• Substitution Method: Another method for solving systems of equations by replacing one variable with an equivalent expression containing the other variable.• Steps: 1. Write an equation containing only one variable, and solve it. 2. Solve for the other variable in either equation. 3. The solution will be an ordered pair. 4. Check to see if the ordered pair makes both equations true.**See example on next slide.
Example of Substitution MethodProblem: y=-4x+8 y=x+7Step 1.Start with one equation. y = -4x+8Substitute x+7 for y. y+7 = -4x+8Solve using Equality Properties. x = 0.2Step 2.Substitute 0.2 for x in y=x+7. y = -4(0.2)+8Simplify. y = 7.2 Solution is (0.2 , 7.2)Step 3. CheckReplace the x and y variable with the solution set. 7.2 - -4 (0.2) + 8 7.2 = 7.2
Solving Using Elimination Method• Elimination Method: Another where you can use the properties of equality to solve a system. You can add of subtract equations to eliminate a variable.• Steps: 1. Look for coefficients that are opposites or each other. If there aren’t any, you may need to multiply one or both equations by a nonzero number. This will to produce coefficients that are opposites of each other. 2. Eliminate one variable. 3. Solve for the remaining variable. 4. Solve for the eliminated variable using the original equations. 5. The solution is an ordered pair. 6. Check to see if the ordered pair makes both equations true.
Example of Elimination MethodProblem: 2x+5y = -22 10x+3y = 22Step 1:Eliminate one variable 5 [ 2x+5y = -22 ] 10x + 25y = -110 10x+3y = 22 10x + 3y = 22 0 + 22y = -132Step 2:Solve for y. 22y = -132 y = -6Step 3:Solve for the eliminated variable using either of the original equations. 2x + 5(-6) = -22 2x – 30 = -22 2x = 8 x=4The solution is (4, -6 )
Additional ResourcesIf you need additional help. Please access the following sites:• The Khan Academy – www.khanacademy.com• Prentice Hall Algebra 1 Textbook Homework Video Tutor – www.PHSchool.com
California Content Standards – 8th Grade Algebra 1• 9.0 Solve a system of two linear equations in two variables algebraically and interpret the answer graphically.• 9.0 Solve a system of two linear equations using three techniques: graphing elimination substitution
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