Answer
There is one solution for these equations: (5, -2), or x = 5, y = -2.
Work Step by Step
Equation 1: $\frac{x + 3}{4} + \frac{y - 1}{3}$ = 1
Equation 2: 2x - y = 12
Let's simplify this first equation. In order to simplify these fractions, we must multiply both sides of the equation by their lowest common denominator, which is 12 for the numbers 3 and 4.
12($\frac{x + 3}{4} + \frac{y - 1}{3}$) = 12(1)
$\frac{12(x + 3)}{4} + \frac{12(y - 1)}{3}$ = 12 (distributive property)
3(x + 3) + 4(y - 1) = 12 (simplify numerator and denominator)
3x + 9 + 4y - 4 = 12 (distributive property)
3x + 4y = 7 (combine like terms and simplify)
Our new 2 equations are:
Equation 1: 3x + 4y = 7
Equation 2: 2x - y = 12
We can easily eliminate the y variable to solve for x by multiplying equation 2 by 4. The +4y and -4y will cancel out in this way.
4(2x - y) = 4(12)
8x - 4y = 48 (distributive property)
Equation 1: 3x + 4y = 7
Equation 2: + 8x - 4y = 48
= 11x + 0y = 55
We can eliminate the 0y because it cancels out. This leaves us with 11x = 55, and dividing both sides by 11 allows us to know that x = 5 (55 / 11 = 5).
To solve for y, plug the x value (5) into either of the original equations.
2(5) - y = 12 (plug in)
10 - y = 12 (subtract 10 from both sides)
-y = 2
y = -2 (divide each side by -1 to make y positive)
The solution is (5, -2), or x = 5, y = -2.
