Question

A system of equations is given.

−2y = 10 − 5x
−3y = −4x +15

Solve for (x, y) using the elimination method. Show all work.

Answer 1

Explanation:

Given the system of equations:

-2y = 10 - 5x

-3y = -4x + 15

Step 1: Multiply both sides of equation (1) by 3 and equation (2) by 2 to make the coefficients of y equal:

3*(-2y) = 3*(10 - 5x)

2*(-3y) = 2*(-4x + 15)

Simplify:

-6y = 30 - 15x

-6y = -8x + 30

Step 2: Add the two equations to eliminate y:

-6y + (-6y) = (30 - 15x) + (-8x + 30)

Combine like terms:

-12y = -23x + 60

Step 3: Divide both sides of the new equation by -12 to solve for x:

x = (-23x + 60) / -12

Distribute the division:

x = (23/12)x - 5

Subtract (23/12)x from both sides:

(12/12)x - (23/12)x = -5

Simplify:

(11/12)x = -5

Divide by -(11/12):

x = (-5) / -(11/12)

x = 60/11

Now that we have found the value of x, we can substitute it back into equation (1) to solve for y:

-2y = 10 - 5x

Substitute x = 60/11:

-2y = 10 - 5 * (60/11)

Simplify:

-2y = 10 - 300/11

Multiply 10 by 11/11 to have a common denominator:

-2y = 110/11 - 300/11

Combine fractions:

-2y = -190/11

Divide by -2:

y = (190/11) / 2

y = 95/11

So, the solution to the system of equations is:

x = 60/11

y = 95/11

Answer 2

(0, - 5 )

Explanation:

- 2y = 10 - 5x → (1)

- 3y = - 4x + 15 → (2)

multiplying (1 ) by 3 and (2) by - 2 and adding will eliminate y

- 6y = 30 - 15x → (3)

6y = 8x - 30 → (4)

add (3) and (4) term by term to eliminate y

(- 6y + 6y) = (- 15x + 8x) + (30 - 30) , that is

0 = - 7x + 0 , that is

- 7x = 0 ⇒ x = 0

substitute x = 0 into either of the 2 original equations and solve for y

substituting into (1)

- 2y = 10 - 5(0)

- 2y = 10 ( divide both sides by - 2 )

y = - 5

solution is (0, - 5 )