Question

solving systems of equations by substitution

1. $y = -\frac{5}{2}x + 10$

$y = \frac{1}{2}x + 4$
solution
( )

1. $-8x + 7y = 10$

$9y = -18$
solution
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1. $3x + 2y = -15$

$2x = -18$
solution
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1. $2x - y = 17$

$x - y = 10$
solution
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1. $-3x + 2y = 23$

$5x + 2y = -17$
solution
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1. $9x - 2y = 19$

$7x = 21$
solution
( )

1. $-2x + y = 10$

$4x - y = -14$
solution
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1. $y = \frac{5}{2}x - 4$

$y = -x + 3$
solution
( )

1. $y = \frac{7}{4}x - 3$

$y = 4$
solution
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1. $y = \frac{2}{3}x - 1$

$y = 3$
solution
( )

Explanation:

Problem 1:

Step1: Set the two equations equal

Since both equations are solved for \( y \), we set them equal: \(-\frac{5}{2}x + 10=\frac{1}{2}x + 4\)

Step2: Combine like terms

Add \(\frac{5}{2}x\) to both sides: \(10=\frac{1}{2}x+\frac{5}{2}x + 4\)
Simplify the right side: \(10 = 3x+4\)
Subtract 4 from both sides: \(6 = 3x\)

Step3: Solve for \( x \)

Divide both sides by 3: \(x = 2\)

Step4: Solve for \( y \)

Substitute \( x = 2 \) into \( y=\frac{1}{2}x + 4\): \(y=\frac{1}{2}(2)+4=1 + 4=5\)

Step1: Solve \( 9y=-18 \) for \( y \)

Divide both sides by 9: \(y=-2\)

Step2: Substitute \( y = -2 \) into \(-8x + 7y = 10\)

\(-8x+7(-2)=10\)
Simplify: \(-8x-14 = 10\)

Step3: Solve for \( x \)

Add 14 to both sides: \(-8x=24\)
Divide by -8: \(x=-3\)

Step1: Solve \( 2x=-18 \) for \( x \)

Divide by 2: \(x=-9\)

Step2: Substitute \( x = -9 \) into \( 3x + 2y=-15 \)

\(3(-9)+2y=-15\)
Simplify: \(-27 + 2y=-15\)

Step3: Solve for \( y \)

Add 27 to both sides: \(2y = 12\)
Divide by 2: \(y = 6\)

Answer:

\((2, 5)\)

Problem 2: