Question

6.3 solve system of equations by linear combination ws#1
7)

$$\begin{cases}-2x + 3y = -2\\\\5x - 10y = 15\\end{cases}$$

8)

$$\begin{cases}5x - 10y = 10\\\\7x - 9y = -6\\end{cases}$$

Explanation:

Step1: Eliminate x for problem 7

Multiply first eq by 5, second by 2:
$5(-2x + 3y) = 5(-2) \implies -10x + 15y = -10$
$2(5x - 10y) = 2(15) \implies 10x - 20y = 30$

Step2: Add equations to solve for y

Sum the two new equations:
$(-10x + 15y) + (10x - 20y) = -10 + 30$
$-5y = 20 \implies y = -4$

Step3: Substitute y to find x (problem7)

Plug $y=-4$ into $5x -10y=15$:
$5x -10(-4) =15 \implies 5x +40=15 \implies 5x=-25 \implies x=-5$

Step4: Eliminate x for problem 8

Multiply first eq by7, second by5:
$7(5x -10y)=7(10) \implies 35x -70y=70$
$5(7x -9y)=5(-6) \implies 35x -45y=-30$

Step5: Subtract equations to solve for y

Subtract second new eq from first:
$(35x -70y)-(35x -45y)=70-(-30)$
$-25y=100 \implies y=-4$

Step6: Substitute y to find x (problem8)

Plug $y=-4$ into $5x -10y=10$:
$5x -10(-4)=10 \implies 5x +40=10 \implies 5x=-30 \implies x=-6$

Answer:

1. $x=-5$, $y=-4$
2. $x=-6$, $y=-4$