Solve x + y = 5, x − y = 1
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = 3, y = 2 |
Step-by-step solution
9 steps-
1 The system, with the equations numbered\begin{aligned}x + y &= 5 & (1)\\ x - y &= 1 & (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 + y = 5 -
3 Subtract y 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 = 5 - yMove the constant terms to the right.
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4 Substitute x = 5 - y into equation (2)
Replacing the unknown by its expression leaves one equation with one unknown, which can be solved on its own.
- y + 5 - y = 1Now there is a single equation in y.
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5 Combine like terms5 - 2 y = 1
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6 Subtract 5 from both sides- 2 y = -4
Move the constant terms to the right.
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7 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{-4}{-2} -
8 Simplifyy = 2
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9 Back-substitute y = 2 to find x
Once one unknown is known, putting its value into an earlier equation gives the other one.
x = 5 - 2 = 3
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | 5 | 5 | ✓ |
| (2) | 1 | 1 | ✓ |
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