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Big Ideas Math: Modeling Real Life, Grade 8

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Big Ideas Math: Modeling Real Life, Grade 8 View details

3. Solving Systems of Linear Equations by Elimination

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Exercise 11 Page 216

The Elimination Method can be used to solve a system of linear equations if either of the variable terms would cancel out the corresponding variable term in the other equation when added together.

(1,-3)

Practice makes perfect

To solve a system of linear equations using the Elimination Method, one of the variable terms needs to be eliminated when one equation is added to or subtracted from the other. This means that either the x- or the y-terms must cancel each other out. 4 x+ 3y=-5 & (I) - x+ 3y=-10 & (II)

We can see that the y-terms will eliminate each other if we subtract Equation (I) from Equation (II).

4x+3y=-5 - x+3y=-10

(II): Subtract (I)

4x+3y=-5 - x+3y-( 4x+3y)=-10-( -5)

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(II):Solve for x

(II): Distribute (-1)

4x+3y=-5 - x+3y-4x-3y=-10-(-5)

(II): a-(- b)=a+b

4x+3y=-5 - x+3y-4x-3y=-10+5

(II): Add and subtract terms

4x+3y=-5 -5x=-5

(II): .LHS /(-5).=.RHS /(-5).

4x+3y=-5 -5x/-5=-5/-5

MovePartNumRight

(II): a* b/c=a/c* b

4x+3y=-5 -5/-5 * x =-5/-5

(II): - a/- b=a/b

4x+3y=-5 5/5 * x =5/5

(II): a/a=1

4x+3y=-5 1 * x = 1

(II): Identity Property of Multiplication

4x+3y=-5 x = 1

Now we can solve for y by substituting the value of x into either equation and simplifying.

4x+3y=-5 x = 1

(I): x= 1

4( 1)+3y=-5 x = 1

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(I):Solve for y

(I): Identity Property of Multiplication

4+3y=-5 x = 1

(I): LHS-4=RHS-4

4+3y-4=-5-4 x = 1

(I): Subtract terms

3y=-9 x = 1

(I): .LHS /3.=.RHS /3.

3y/3=-9/3 x=1

MovePartNumRight

(I): a* b/c=a/c* b

3/3 * y =-9/3 x=1

(I): a/a=1

1 * y =-9/3 x=1

(I): Identity Property of Multiplication

y=-9/3 x=1

MoveNegNumToFrac

(I): Put minus sign in front of fraction

y=-9/3 x=1

(I): Calculate quotient

y=-3 x=1

The solution, or point of intersection, of the system of equations is (1,-3). To check this solution, we will substitute it back into the given system and simplify. If doing so results in true statements for every equation in the system, our solution is correct.

4x+3y=-5 - x+3y=-10

(I), (II): x= 1, y= -3

4( 1)+3( -3) ? = -5 - 1+3( -3) ? = -10

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Simplify

(I), (II): a(- b)=- a * b

4(1)+(-3 * 3) ? = -5 -1+(-3 * 3) ? = -10

(I), (II): Multiply

4(1)+(-9) ? = -5 -1+(-9) ? = -10

(I): Identity Property of Multiplication

4+(-9) ? = -5 -1+(-9) ? = -10

(I), (II): a+(- b)=a-b

4-9 ? = -5 -1-9 ? = -10

(I), (II): Subtract terms

-5 = -5 ✓ -10 = -10 ✓

Subchapter links

Try It p.212-214

Self-Assessment for Concepts & Skills p.214

Self-Assessment for Problem Solving p.215

Review & Refresh p.216

Concepts, Skills, & Problem Solving p.216-218