005 Sample Final A, Question 9

Question Solve the following system of equations

${\begin{aligned}2x+3y&=&1\\-x+y&=&-3\end{aligned}}$

Foundations:
1) What are the two methods for solving a system of equations?
2) How do we use the substitution method?
3) How do we use the elimination method?
Answer:
1) The two methods are the substitution and elimination methods.
2) Solve for x or y in one of the equations and substitute that value into the other equation.
3) Multiply one equation by some number on both sides to make one of the variables, x or y, have the same coefficient and add the equations together.

Step 1:
Add two times the second equation to the first equation. So we are adding $-2x+2y=-6$ to the first equation.
This leads to:
${\begin{array}{rcl}0+5y&=&-5\\-x+y&=&-3\end{array}}$

Step 2:
This gives us that $y=-1.$
Now we just need to find x. So we plug in -1 for y in the second equation.
${\begin{array}{rcl}-x-1&=&-3\\-x&=&-2\\x&=&2\end{array}}$

Final Answer:
$x=2,~y=-1$