Solve 5x + y = −29, −x − y = 9 by substitution
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = −5, y = −4 |
Step-by-step solution
10 steps-
1 The system, with the equations numbered\begin{aligned}5 x + y &= -29 & (1)\\ - x - y &= 9 & (2)\end{aligned}
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2 Solve equation (1) 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.
5 x + y = -29 -
3 Subtract 5x 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 = - 5 x - 29Move the constant terms to the right.
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4 Substitute y = -5x - 29 into equation (2)
Replacing the unknown by its expression leaves one equation with one unknown, which can be solved on its own.
- x - - 5 x - 29 = 9Now there is a single equation in x.
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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.
- x + 5 x + 29 = 9A minus sign in front of a bracket changes the sign of every term inside.
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6 Combine like terms4 x + 29 = 9
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7 Subtract 29 from both sides4 x = -20
Move the constant terms to the right.
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8 Divide both sides by 4
Dividing both sides by 4 undoes the multiplication by 4, 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{-20}{4} -
9 Simplifyx = -5
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10 Back-substitute x = -5 to find y
Once one unknown is known, putting its value into an earlier equation gives the other one.
y = -29 - 5 \left(-5\right) = -4
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | −29 | −29 | ✓ |
| (2) | 9 | 9 | ✓ |
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