Solve 2x − 3y = −28, −x − y = −1 by elimination
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = −5, y = 6 |
Step-by-step solution
8 steps-
1 The system, with the equations numbered\begin{aligned}2 x - 3 y &= -28 & (1)\\ - x - y &= -1 & (2)\end{aligned}
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2 Multiply equation (2) by 2 so the x-terms match
Elimination needs one unknown to have the same or opposite coefficient in both equations. Multiplying a whole equation by a number keeps it true, so the equations are scaled until the x-terms match.
\begin{aligned}2 x - 3 y &= -28 & (1)\\ - 2 x - 2 y &= -2 & (2)\end{aligned} -
3 Add the two equations to eliminate x
Adding or subtracting two true equations gives another true equation. Because the x-terms match, they cancel, leaving one equation in one unknown.
- 5 y = -30 -
4 Divide both sides by -5
Dividing both sides by -5 undoes the multiplication by -5, 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{-30}{-5} -
5 Simplifyy = 6
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6 Substitute y = 6 into equation (2)
Once one unknown is known, putting its value into an earlier equation gives the other one.
- x - 6 = -1 -
7 Add 6 to 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 = 5Move the constant terms to the right.
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8 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.
x = -5
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | −28 | −28 | ✓ |
| (2) | −1 | −1 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.