Solve −2x + 3y = 12, 4x − 4y = −12 by substitution
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = 3, y = 6 |
Step-by-step solution
12 steps-
1 The system, with the equations numbered\begin{aligned}3 y - 2 x &= 12 & (1)\\ 4 x - 4 y &= -12 & (2)\end{aligned}
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2 Solve equation (1) for x
Substitution starts by solving one equation for one unknown, ideally one with coefficient 1 or −1. Then x can be replaced in the other equation.
3 y - 2 x = 12 -
3 Subtract 3y from both sides
An equation stays true as long as you do the same thing to both sides, like adding the same weight to both pans of a balance. Adding or subtracting the same amount on both sides moves a term to the other side, where it appears with the opposite sign.
- 2 x = 12 - 3 yMove the constant terms to the right.
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4 Divide both sides by -2
Dividing both sides by -2 undoes the multiplication by -2, so the variable is left on its own. Both sides change in the same way, so the equation stays true (dividing by 0 is the one thing that is never allowed).
x = \frac{12 - 3 y}{-2} -
5 Simplifyx = \frac{3 y}{2} - 6
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6 Substitute x = 3y/2 - 6 into equation (2)
Replacing the unknown by its expression leaves one equation with one unknown, which can be solved on its own.
- 4 y + 4 \left(\frac{3 y}{2} - 6\right) = -12Now there is a single equation in y.
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7 Distribute: multiply each term inside the brackets
The distributive property, a(b + c) = ab + ac: the number in front of a bracket multiplies every term inside it, not just the first. A minus in front works like −1, so it flips the sign of every term inside.
- 4 y + 6 y - 24 = -12 -
8 Combine like terms2 y - 24 = -12
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9 Add 24 to both sides2 y = 12
Move the constant terms to the right.
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10 Divide both sides by 2
Dividing both sides by 2 undoes the multiplication by 2, so the variable is left on its own. Both sides change in the same way, so the equation stays true (dividing by 0 is the one thing that is never allowed).
y = \frac{12}{2} -
11 Simplifyy = 6
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12 Back-substitute y = 6 to find x
Once one unknown is known, putting its value into an earlier equation gives the other one.
x = \frac{3 \cdot 6}{2} - 6 = 3
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | 12 | 12 | ✓ |
| (2) | −12 | −12 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.