### Home > INT3 > Chapter 12 > Lesson 12.1.1 > Problem12-10

12-10.

Use the algebraic strategies you have developed to solve the system of equations below. Be sure to check your solution.

$2x + y – 3z = –12$
$5x – y + z = 11$
$x + 3y – 2z = –13$

Choose a pair of equations that can be added together to eliminate a variable.

 $\left. \begin{array}{l}{ 2 x + y - 3 z = - 12 }\\{ + 5 x - y + z = 11 }\\\end{array} \right.$ ${ 7 x \ \ \ \ \ \ \ \ \ \ - 2 z = - 1 }$

Choose a different pair of equations to add together to eliminate the same variable.
In this case, you will have to multiply each equation by a constant.

 $\left. \begin{array}{l}{-3( 2 x + y - 3 z = - 12) }\\{ + \ \ \ \ \ x + 3y -2z = -13 }\\\end{array} \right.$ $-5x \ \ \ \ \ \ \ \ \ +7z=23$

Solve the system of equations you created in steps 1 and 2 for either $x$ or $z$.

Solve for $x$ using the two-variable equation you created in either step 1 or step 2.
Then substitute the values for $x$ and $z$ into one of the original equations to solve for $y$.

Check your solution in a different equation than the one you used to solve for $y$.