Solve 2x + 3y = 12, 4x − y = 10
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = 3, y = 2 |
Step-by-step solution
11 steps-
1 The system, with the equations numbered\begin{aligned}2 x + 3 y &= 12 & (1)\\ 4 x - y &= 10 & (2)\end{aligned}
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2 Solve equation (2) for y
Substitution starts by solving one equation for one unknown, ideally one with coefficient 1 or −1. Then y can be replaced in the other equation.
4 x - y = 10 -
3 Subtract 4x 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.
- y = 10 - 4 xMove the constant terms to the right.
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4 Multiply both sides by −1
Multiplying both sides by −1 flips every sign. The equation stays true, and a positive variable or leading term is easier to read and to factor.
y = 4 x - 10 -
5 Substitute y = 4x - 10 into equation (1)
Replacing the unknown by its expression leaves one equation with one unknown, which can be solved on its own.
2 x + 3 \left(4 x - 10\right) = 12Now there is a single equation in x.
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6 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.
2 x + 12 x - 30 = 12 -
7 Combine like terms14 x - 30 = 12
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8 Add 30 to both sides14 x = 42
Move the constant terms to the right.
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9 Divide both sides by 14
Dividing both sides by 14 undoes the multiplication by 14, 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{42}{14} -
10 Simplifyx = 3
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11 Back-substitute x = 3 to find y
Once one unknown is known, putting its value into an earlier equation gives the other one.
y = 4 \cdot 3 - 10 = 2
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | 12 | 12 | ✓ |
| (2) | 10 | 10 | ✓ |
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