Substitution Method — Definition & Examples

Swap them out!

When you have a problem with at least one variable, you can swap the variable out with a number in order to learn more about the problem. For example, take the expression $y = 2x +5$.

The variables are $x$ and $y$. If I then ask you to find out what $y$ is when $x = 3,$ you can replace $x$ with $3$ in the expression: $$y = 2(3) + 5$$

Then, you can multiply and add to find out what $y$ is when $x$ is $3.$ Let's try the same problem with $x$ as a different number:

When $x = 5$, $y$ is $15$. Here are the steps I took to figure that out:

1. Swap $5$ for $x$ in the expression: $y = 2(5) + 5$
2. Multiply: $y = 10 + 5$
3. Add: $y = 15$

(Video) Evaluating Variable Expressions

by mathman1024

Here's a video by mathman1024 that shows you how to evaluate variable expressions.

$\large{\text{Summary}}$

Let's start off with an example: $4x^2-5y$ for $x=-2$, $y=-11$. In order to evaluate this expression we just plug the numbers into the expression where their corresponding variables are. Make sure you use parentheses in order to avoid mistakes. This makes it easier to apply the order of operations as well. $$\begin{align} &4x^2-5y\\ &=4{\color{#8925AE}{(-2)}}^2-5{\color{#29A603}{(-11)}}\\ &=4(4)-5(-11)\\ &=16-(-55)\\ &=71 \end{align}$$

Try this next one on your own and then watch the video to see the answer!

1. $-3xy^3+4+7y\ \text{ when }x=9,\ y=-1$

Substituting for a Variable

Let's say you have an expression with a variable in it. If you know the value that the variable is equal to, you can substitute that value in for the variable in the expression! Let's look at an example.

Since we know that $x=5,$ we can directly substitute or replace the $x$ in the expression $x+3$ with a $5$ to solve for $y!$

This is the reverse of solving an equation. When solving an equation you are trying to find the value of the variable. Here, we've given it to you.

Let's do another example.

The question is asking us to evaluate $u=6v+5$ when $v=3.$ This means that wherever we see a $v,$ we can substitute in a $3.$ $$\begin{align} u&=6v+5\\ u&=6(3)+5\\ u&=18+5\\ u&=23 \end{align}$$