Question

What is the solution to the system of equations?

Answer 1

Call the two equations above A and B, in order to not confuse them.

A: -2x + 6y = -38

B: 3x - 4y = 32

For this system we have opposites in x and y, so Elimination (or Linear Combination) works best. Either variable works, so let's work with x first and multiply A by 3 and B by 2. This is done so we get opposites in A and B then when added together give zero.

-2x + 6y = -38 ------> multiply by 3 ----> -6x + 18y = -114

3x - 4y = 32 ------> multiply by 2 -----> 6x - 8y = 64

Now we add the new equations. The -6x and 6x are opposites and go away. We are left with

10y = -50. We divide both sides by 10 and get that y = -5.

Now we take y = -5 and put it into an original equation. Let's use A.

-2x + 6y = -38 the original equation A

-2x + 6(-5) = -38 we found y = -5

-2x + (-30) = -38 evaluating and multiplying

-2x - 30 = -38 apply the parentheses

-2x = -8 add 30 to both sides

x = 4 divide on both sides by -2

Thus x = 4 and y = -5, or (4, -5) is the solution.

Answer 2

$\left \{ {{-2x+6y=-38} \atop {3x-4y=32 }} \right.\\ In this case we need to find the Least Common Factor with the coefficients in front of the "x" or the "y". The coefficients in front of the "x" in both equations are smaller so we will deal with these because it is easier for us. The Least Common Factor of -2 and 3 is 6. \\Now we need to find what these numbers multiply by to turn into 6. The -2 multiplies by 3 to turn into -6 and the 3 multiplies by 2 to turn into 6.$

The reason we find the Least Common Factor for one of the variables is to have the coefficients from the top and bottom cancel out each other so we need to make sure one of them is negative and the other is positive.

Now we multiply the entire top equation by the number we multiplied the coefficient infront of x to get -6, and that number is "3" ; so multiplying the rest would by 6 * 3 and -38 * 3. We do the same with the bottom equation but we multiply the rest of the numbers by "2" because that is what we multiplied the 3 by. We would get:

$\left \{ {{-6x+18y=-114} \atop {6x-8y=64}} \right.$

Now we add the values of each coefficient with the ones underneath that correspond to the same variable and we get:

$10y=-50\\This is because -6x + 6x =0 , 18y-8y = 10y , and -114 +64 = -50\\We then solve for y and get y=-5$

Now we plug in y=-5 into one of the equations and we get:

3x - 4(-5) =32 .... 3x +20 =32 .... 3x=12 ... x=4

x=4 and y= -5