Solve 2x − y = 19, x − 5y = 32
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = 7, y = −5 |
Step-by-step solution
11 steps-
1 The system, with the equations numbered\begin{aligned}2 x - y &= 19 & (1)\\ x - 5 y &= 32 & (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.
2 x - y = 19 -
3 Subtract 2x 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 = 19 - 2 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 = 2 x - 19 -
5 Substitute y = 2x - 19 into equation (2)
Replacing the unknown by its expression leaves one equation with one unknown, which can be solved on its own.
x - 5 \left(2 x - 19\right) = 32Now 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.
x - 10 x + 95 = 32A minus sign in front of a bracket changes the sign of every term inside.
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7 Combine like terms95 - 9 x = 32
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8 Subtract 95 from both sides- 9 x = -63
Move the constant terms to the right.
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9 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).
x = \frac{-63}{-9} -
10 Simplifyx = 7
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11 Back-substitute x = 7 to find y
Once one unknown is known, putting its value into an earlier equation gives the other one.
y = 2 \cdot 7 - 19 = -5
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | 19 | 19 | ✓ |
| (2) | 32 | 32 | ✓ |
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