Solve −x − y = −7, −3x + 3y = −3 by elimination
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = 4, y = 3 |
Step-by-step solution
8 steps-
1 The system, with the equations numbered\begin{aligned}- x - y &= -7 & (1)\\ 3 y - 3 x &= -3 & (2)\end{aligned}
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2 Multiply equation (1) by 3 so the y-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 y-terms match.
\begin{aligned}- 3 x - 3 y &= -21 & (1)\\ 3 y - 3 x &= -3 & (2)\end{aligned} -
3 Add the two equations to eliminate y
Adding or subtracting two true equations gives another true equation. Because the y-terms match, they cancel, leaving one equation in one unknown.
- 6 x = -24 -
4 Divide both sides by -6
Dividing both sides by -6 undoes the multiplication by -6, 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{-24}{-6} -
5 Simplifyx = 4
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6 Substitute x = 4 into equation (1)
Once one unknown is known, putting its value into an earlier equation gives the other one.
- y - 4 = -7 -
7 Add 4 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.
- y = -3Move 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.
y = 3
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | −7 | −7 | ✓ |
| (2) | −3 | −3 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.