# Solving Systems of Equations by Elimination

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Solving Systems of Equations by Elimination
Solving by elimination can be done by addition or multiplication.

Solving Systems of Equations by Elimination
Solve by addition Ex) 3x – 5y = -16 2x + 5y = 31 Notice that there is an inverse here (-5y and 5y)

Solving Systems of Equations by Elimination
3x – 5y = -16 the -5y and 5y will cancel +2x + 5y = 31 add like terms 5x + 0 = 15 divide by 5 x = 3 Now substitute the 3 into either equation for x and solve for y. 3(3) – 5y = – 5y = -16 solve equation y = -25 y = 5 solution (3,5)

Solving Systems of Equations by Elimination
Solve by multiplication If there is not an inverse, you need to multiply one or both of the equations to make an inverse in the problem. Ex) 5x + 2y = 6 9x + 2y = 22

Solving Systems of Equations by Elimination
Solve by multiplication If there is not an inverse, you need to multiply one or both of the equations to make an inverse in the problem. Ex) 5x + 2y = x + 2y = 6 9x + 2y = (9x + 2y = 22) Now eliminate the 2y and -2y

Solving Systems of Equations by Elimination
Solve by multiplication If there is not an inverse, you need to multiply one or both of the equations to make an inverse in the problem. Ex) 5x + 2y = x + 2y = 6 -1(9x + 2y = 22) x – 2y = -22 Now eliminate the 2y and -2y

Solving Systems of Equations by Elimination
Ex) 5x + 2y = 6 -9x – 2y = x = -16 x = 4 Now substitute the 4 in for x and solve for y 5(4) + 2y = y = 6 2y = -14 y = -7 solution (4, -7)

Solving Systems of Equations by Elimination
Ex) 3x + 4y = 6 5x + 2y = -4

Solving Systems of Equations by Elimination
Ex) 3x + 4y = 6 5x + 2y = -4 Sometimes, there is nothing obvious to inverse, you may have to multiply one or both equations by a number to inverse. 3x + 4y = 6 3x + 4y = 6 -2(5x + 2y = 4) -10x – 4y = 8

Solving Systems of Equations by Elimination
3x + 4y = 6 -10x – 4y = 8 -7x = 14 x = -2 3(-2) + 4y = y = 6 4y = 12 y = 3 solution (-2,3)

Solving Systems of Equations by Elimination
Ex) -3x – 3y = x + 8y = 16

Solving Systems of Equations by Elimination
Ex) -3x – 3y = x + 8y = 16 When neither equation has anything in common, you will have to multiply BOTH equations to find an inverse. -2(-3x – 3y = -21) 6x + 6y = 42 3(-2x + 8y = 16) -6x + 24y = 48

Solving Systems of Equations by Elimination
6x + 6y = 42 -6x + 24y = 48 30y = 90 y = 3 6x + 6(3) = 42 6x + 18 = 42 6x = 24 x = 4 solution (4, 3)

Solving Systems of Equations
Ex) 3x – 6y = 10 x – 2y = 4

Solving Systems of Equations with No Solution
Ex) 3x – 6y = 10 x – 2y = 4 Make inverse 3x – 6y = 10 3x – 6y = 10 -3(x – 2y = 4) -3x + 6y = = -2 0 ≠ -2 therefore, there is no solution

Solving Systems of Equations
Ex) 3x + 6y = 24 -2x – 4y = -16

Solving Systems of Equations with Infinite Solutions
Ex) 3x + 6y = 24 -2x – 4y = -16 Find the inverse 2(3x + 6y = 24) 6x + 12y = 48 3(-2x – 4y = -16) -6x – 12y = = 0 0 = 0 is a true statement, there are infinite solutions. They would graph as the same line.

Solving Systems of Equations
Method Best time to use Graphing *to estimate a solution Substitution *when one variable has a coefficient of 1 or -1 Elimination *when one of the variables has the same or opposite coefficients. *when there are no other options for solving.