Solve 3x + 2y = −5, 2x − 5y = 3 by substitution
Worked out line by line the way a teacher would write it.
Answer
| Solution | x = −1, y = −1 |
Step-by-step solution
13 steps-
1 The system, with the equations numbered\begin{aligned}3 x + 2 y &= -5 & (1)\\ 2 x - 5 y &= 3 & (2)\end{aligned}
-
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.
3 x + 2 y = -5 -
3 Subtract 2y 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.
3 x = - 2 y - 5Move the constant terms to the right.
-
4 Divide both sides by 3
Dividing both sides by 3 undoes the multiplication by 3, 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{- 2 y - 5}{3} -
5 Simplifyx = - \frac{2 y}{3} - \frac{5}{3}
-
6 Substitute x = -2y/3 - 5/3 into equation (2)
Replacing the unknown by its expression leaves one equation with one unknown, which can be solved on its own.
- 5 y + 2 \left(- \frac{2 y}{3} - \frac{5}{3}\right) = 3Now there is a single equation in y.
-
7 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.
- 5 y - \frac{4 y}{3} - \frac{10}{3} = 3 -
8 Combine like terms- \frac{19 y}{3} - \frac{10}{3} = 3
-
9 Multiply both sides by 3 to clear the fractions
3 is the least common multiple of the denominators. Multiplying every term on both sides by it cancels each fraction, so the rest of the work uses whole numbers. The equation stays balanced because both sides were multiplied by the same number.
- 19 y - 10 = 9 -
10 Add 10 to both sides- 19 y = 19
Move the constant terms to the right.
-
11 Divide both sides by -19
Dividing both sides by -19 undoes the multiplication by -19, 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{19}{-19} -
12 Simplifyy = -1
-
13 Back-substitute y = -1 to find x
Once one unknown is known, putting its value into an earlier equation gives the other one.
x = - \frac{5}{3} - \frac{2 \left(-1\right)}{3} = -1
Check by substitution
| Equation | Left side | Right side | |
|---|---|---|---|
| (1) | −5 | −5 | ✓ |
| (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.