# What is the final step of completing a solve by substitution problem?

Aug 20, 2016

I'm not sure where exactly you mean "final" in the solving process. So, I've prepared a couple of problems that I will work through slowly and carefully, showing all the steps to the final answer.

Example 1: Solve the following system of equations-$2 x + y = 5 , 3 x + 2 y = 9$

Since we want to solve with substitution, we must solve for one variable in one of the equations. I think it would be easiest to solve for $y$ in the first equation.

$y = 5 - 2 x$

We can now substitute into the other equation:

$3 x + 2 \left(5 - 2 x\right) = 9$

$3 x + 10 - 4 x = 9$

$- x = - 1$

$x = 1$

We must now find the value of $y$. This is found by inserting $x = 1$ into one of the equations and solving for $y$.

$y = 5 - 2 x$

$y = 5 - 2 \left(1\right)$

$y = 3$

Hence, our solution set is $\left\{1 , 3\right\}$.

Example 2: Find all real values of $x$ and $y$ that satisfy the following system of equations: $3 y = - 2 {x}^{2} + 2 , 2 {x}^{2} - 3 {y}^{2} = - 4$

Once again, as with the last example, we need to solve for one of the variables in one of the equations. It looks easiest to isolate the $y$ in the first equation, however solving this equation won't be as neat as solving the previous one.

$y = - \frac{2}{3} {x}^{2} + \frac{2}{3}$

We can now substitute into equation #2.

$2 {x}^{2} - 3 {\left(- \frac{2}{3} {x}^{2} + \frac{2}{3}\right)}^{2} = - 4$

$2 {x}^{2} - 3 \left(\frac{4}{9} {x}^{4} - \frac{8}{9} {x}^{2} + \frac{4}{9}\right) = - 4$

$2 {x}^{2} - \frac{4}{3} {x}^{4} + \frac{8}{3} {x}^{2} - \frac{4}{3} = - 4$

$- \frac{4}{3} {x}^{4} + \frac{14}{3} {x}^{2} + \frac{8}{3} = 0$

Solve using a graphing calculator. If it's a standard one, like a TI84, use ${y}_{1} = - \frac{4}{3} {x}^{4} + \frac{14}{3} {x}^{2} + \frac{8}{3} = 0$ and ${y}_{2} = 0$, and press "calc" followed by "intersect".

This will give you real roots of $2$ and $- 2$. All that is left to do is solve for $y$.

$y = - \frac{2}{3} {\left(2\right)}^{2} + \frac{2}{3} \text{ AND } - \frac{2}{3} {\left(- 2\right)}^{2} + \frac{2}{3}$

$y = - \frac{8}{3} + \frac{2}{3} \text{ AND } - \frac{8}{3} + \frac{2}{3}$

$y = - 2 \text{ AND } - 2$

Hence, our solution set is $\left\{2 , - 2\right\}$ and $\left\{- 2 , - 2\right\}$.

Use the following practice exercises to develop your comfort with the skills dealt with in this answer.

Practice exercises:

1. Find the real values of $x$ and $y$ that satisfy the following systems of equations.

a) $2 x - 3 y = 4 , x + 2 y = 9$

b) $3 x + y = - 2 , {x}^{2} = y$

c) $2 {x}^{2} - 3 {y}^{2} = - 10 , {x}^{2} - 2 x + 3 y = 5$

Hopefully this helps, and good luck!