Question

9-9. solve each system of equations below. homework help
a. $2x + y = -7y$
$y = x + 10$
b. $3x = -5y$
$6x - 7y = 17$

Explanation:

Part a

Step1: Simplify the first equation

We have the equation \(2x + y=-7y\). Let's move all \(y\) terms to one side. Add \(7y\) to both sides:
\(2x + y+ 7y=-7y + 7y\)
\(2x+8y = 0\) \(\quad(1)\)
The second equation is \(y=x + 10\) \(\quad(2)\)

Step2: Substitute equation (2) into equation (1)

Substitute \(y=x + 10\) into \(2x+8y=0\):
\(2x+8(x + 10)=0\)
Expand the left - hand side: \(2x+8x+80 = 0\)
Combine like terms: \(10x+80=0\)

Step3: Solve for \(x\)

Subtract 80 from both sides: \(10x=0 - 80=-80\)
Divide both sides by 10: \(x=\frac{-80}{10}=-8\)

Step4: Solve for \(y\)

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

Part b

Step1: Express \(x\) from the first equation

We have the equation \(3x=-5y\), so \(x=\frac{-5y}{3}\) \(\quad(1)\)
The second equation is \(6x-7y = 17\) \(\quad(2)\)

Step2: Substitute equation (1) into equation (2)

Substitute \(x=\frac{-5y}{3}\) into \(6x-7y = 17\):
\(6\times\frac{-5y}{3}-7y=17\)
Simplify the first term: \(2\times(-5y)-7y = 17\)
\(-10y-7y=17\)

Step3: Solve for \(y\)

Combine like terms: \(-17y=17\)
Divide both sides by \(-17\): \(y=\frac{17}{-17}=-1\)

Step4: Solve for \(x\)

Substitute \(y = - 1\) into equation (1): \(x=\frac{-5\times(-1)}{3}=\frac{5}{3}\)

Answer:

a. \(x=-8,y = 2\)

b. \(x=\frac{5}{3},y=-1\)