Solve x + 2y = 10, −5x − y = 13
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = −4, y = 7 |
Step-by-step solution
10 steps-
1 The system, with the equations numbered\begin{aligned}x + 2 y &= 10 & (1)\\ - 5 x - y &= 13 & (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.
x + 2 y = 10 -
3 Subtract 2y 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.
x = 10 - 2 yMove the constant terms to the right.
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4 Substitute x = 10 - 2y into equation (2)
Replacing the unknown by its expression leaves one equation with one unknown, which can be solved on its own.
- y - 5 \left(10 - 2 y\right) = 13Now there is a single equation in y.
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5 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.
- y - 50 + 10 y = 13A minus sign in front of a bracket changes the sign of every term inside.
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6 Combine like terms9 y - 50 = 13
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7 Add 50 to both sides9 y = 63
Move the constant terms to the right.
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8 Divide both sides by 9
Dividing both sides by 9 undoes the multiplication by 9, 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{63}{9} -
9 Simplifyy = 7
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10 Back-substitute y = 7 to find x
Once one unknown is known, putting its value into an earlier equation gives the other one.
x = 10 - 2 \cdot 7 = -4
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | 10 | 10 | ✓ |
| (2) | 13 | 13 | ✓ |
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