How would I solve this system using substitution?

Question

-8x+7y=-23

6x-5y=15

Answer 1

A more in-depth explanation of my solving process:

With the substitution method, you are solving for one of the variables in one of the equations. It doesn't matter which equation you use to solve for the variable you are going to substitute. I generally always solve for y first and I pick the easiest looking equation to solve for y to start with. After solving for y in the chosen equation I would plug what I got for y which will still have an x in it and substitute that into the equation I haven't used yet. Because we solve for y in the "first" equation when we plug that answer into the "second" equation we now only have one variable x. We then proceed to solve for x. Once we know what x equals we can plug that into either equation to find what y equals.

The step-by-step version of solving the equation with substitution:

Question:

-8x+7y=-23

6x-5y=15

Step 1. Choose which equation and variable you want to use.

(Here I chose to solve for y in the second equation)

-5y=-6x+15

y=(6/5)x-3

Step 2. Substitute answer into other equations

-8x+7[(6/5)x-3]=-23

Step 3. Solve for your given variable

-8x+(42/5)x-21=-23

-(40/5)x+(42/5)x=-2

(2/5)x=-2

x=-5

Step 4. Find the second variable by plugging in the answer you got for the first variable into one of your original equations.

(Here I chose to use the second equations again)

6(-5)-5y=15

-30-5y=15

-5y=45

y=-9

Answer: x= -5 and y= -9

Similarly, you could have solved for x first then used the same method to find what y equals, and then go back to find out what x equals. Hopefully, this helps answer your question!

Answer 2

starts with 6x-5y = 15

6x = 5y + 15

= 5(y+3)

so then x = (5/6)(y+3)

Next, multiplies the first equation by -1:

8x - 7y = 23

Substitutes:

8 * 5/6(y+3) - 7y = 23

(40/6)(y+3) - 7y = 23 <-- multiplies the fractions

(20/3)(y+3) - 7y = 23 <---redduces

(20/3)(y+3) = 7y + 23 <-- adds 7y to both sides

y+3 = (3/20)(7y+23) <--- multiplies both sides by 3/20

(20/20) y + 60/20 = (21/20)y + 69/20 <--- common denominator is 20

60/20 = (1/20)y + 69/20 <--- subtracts y = 1*y = (20/20)y from both sides

0 = (1/20)y + 9/20 <--- subtracts 3=60/20 from both sides

0 = y+9 <--- multiplies both sides by 20 to kill the fractions

y = -9

Then x = (5/6)(-9 + 3)

= (5/6)(-6)

= -5

The apparent solution is (x=-5, y=-9)

checks

========

1st equation: (-8)(-5) + 7(-9) = 40 - 63 = -23

yes, it checks!

2nd equation: 6(-5) - 5(-9)

= -30 - (-45)

= 15

yes it checks..

The OFFICIAL solution is (x=-5, y=-9)