# Question #9364f

Feb 7, 2017

I won't solve all of them, but I will explain the method and work a couple of examples. Read on...

#### Explanation:

The number in front of each variable is called its coefficient. One method (my favourite) is to multiply one of the equations (meaning, multiply every coefficient and the constant term) by whatever number necessary to make the coefficients equal but opposite in sign for a given variable.

$5 x + 2 y = 29$
$x - y = - 4$

Multiply the entire second equation by 2, so that the coefficients of the $x$-terms match:

$5 x + 2 y = 29$
$2 x - 2 y = - 8$

Now, add each term in the second equation to the like term in the first one:

$5 x + 2 x = 7 x$
$2 y + \left(- 2 y\right) = 0$ (this is what you need to happen!)
$29 + \left(- 8\right) = 21$

Put these terms (on the right side above) together, much like they were in the starting equations:

$7 x = 21$

$x = 3$

Now, put $x = 3$ into either of the starting equations to find the value of $y$

$x - y = - 4$ becomes $3 - y = - 4$

$y = 7$

(You can test using the other equation, just to be sure.)

For example 2, multiply the second equation by -1:

$2 x - y = - 1$
$- 3 x + y = - 2$

Add: $2 x + \left(- 3 x\right) = - 1 x$
$- y + y = 0$
$- 1 + \left(- 2\right) = - 3$

Altogether: $- 1 x = - 3$ $x = 3$

Substitute into the first equation: $2 \left(3\right) - y = - 1$
$6 - y = - 1$
$- y = - 1 - 6$
$y = 7$