Associated Topics || Dr. Math Home || Search Dr. Math

### Two Equations with Two Unknowns

```
Date: 05/24/99 at 01:00:47
From: Margie Yeager
Subject: Algebra 2

1)  2x-3y = 50
7x+8y = -10

2)  5x+6y = 16
3x-4y = 2

3)  3x+2y = 14
2x+3y = 1

4)  2x+3y = 10
5x-4y = 2

5)  10x+3y = 1
3x-2y = -10

6)  5x+2y = 9
2x+3y = -3
```

```
Date: 05/24/99 at 18:03:00
From: Doctor Rick
Subject: Re: Algebra 2

Hi, Margie. I will do the first of your problems to show you the
technique.

Before you add or subtract, you want to multiply one or both equations
by constants (that is, multiply both sides of one equation by the same
number). Your goal is to eliminate one variable when you do the

Let's eliminate the y. To do this, we want to have the coefficient of
y in the two equations be either the same or the opposite. One way to
do this is to multiply each equation by the coefficient of y in the
OTHER equation.

Multiply the first equation on both sides by 8:

16x - 24y = 400

Multiply the second equation on both sides by 3:

21x + 24y = -30

Now the coefficients of y are opposite, so we can add the two
equations:

16x - 24y = 400
21x + 24y = -30
---------------
37x +  0y = 370

The y term has vanished:

37x = 370

Divide both sides by 37:

x = 370/37 = 10

We aren't done yet because we only have a value for x, not for y.
Choose either of the original equations, substitute x = 10, then solve
for y:

2x - 3y = 50
2(10) - 3y = 50
20 - 3y = 50
-3y = 50 - 20 = 30
y = 30/(-3) = -10

The solution set is { x = 10, y = -10 }.

Last but not least, check the answer by substituting these values in
both equations:

2x - 3y = 50              7x + 8y = -10
2(10) - 3(-10) = 50       7(10) + 8(-10) = -10
20 + 30 = 50              70 - 80 = -10
50 = 50                   -10 = -10

We're done! Now it's your turn.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Linear Equations

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search