Substituting values

Expressions vs equations

I’ll start this lesson by quickly explaining some math terminology. You may have heard or read phrases like these before: “Simplify the expression” and “Solve the equation for x.” Well, what is the difference between an expression and an equation?

It’s simple: equations have equals signs, and expressions don’t. Or, more formally: an equation is composed of two mathematical expressions joined by an equals sign.

$ 2(x+1)+1 $ is an expression. Given this expression by itself it is not possible to solve for $ x $. It would be possible to simplify the way the expression is written like so: $ 2(x+1)+1=2x+2+1=2x+3 $.

$ 11 $ is also an expression. We can create an equation out of these two expressions by joining them with an equals sign: $ 2x+3=11 $. Now we have something solvable, using some simple algebra we can solve this equation for $ x $:

$\begin{align*} & 2x+3=11 & \text{subtract 3 from both sides} \\ & \rightarrow2x=8 & \text{divide both sides by 2} \\ & \rightarrow x=4 \end{align*}$

Solving equations like we just did will be the focus of the Linear Equations In One Variable lesson. In this lesson we will focus only on substituting values into expressions, as explained below.

Substituting values into expressions

Some questions ask you to substitute numerical values in for the variables of an expression. For example:

$\text{If x=2, what is the value of } \frac{x^{3}-x}{x-4}\text{?}$

To compute the value of the expression we would insert 2 into the expression everywhere an x appears, then we would simplify the expression by multiplying and adding the terms.

$\frac{2^{3}-2}{2-4}=\frac{8-2}{2-4}=\frac{6}{-2}=-3$

So, for $ x=2 $, the expression $ \frac{x^{3}-x}{x-4} $ has a value of $ -3 $.

Here is different type of example:

$\text{If } x=2\cdot a+1\text{, and }a=3\text{, solve for x}$

Now to be clear, this is an equation. But, because $ x $ is already isolated on the left hand side, to solve for $ x $ all we need to do is substitute $ a=3 $ into the expression $ 2 \cdot a+1 $. Doing so we can easily solve that $ x=7 $.

Here’s an example where we must substitute two values:

$\text{If }a=9\text{, and }b=5\text{, simplify }\frac{a+b}{a-b}$

The question asks us to substitute $ 9 $ for $ a $ and $ 5 $ for $ b $, then simplify the fraction:

$\frac{9+5}{9-5}=\frac{14}{4}=\frac{7}{2}$

Word problems can be a bit trickier because you have to translate the words in an equation to solve, but once you get there the math is the same. Here’s an example:

A school district uses the following equation to calculate the salary paid to a teacher:

$\text{Salary }=30,000+1,000Yp-500Yf$

Where $ Yp $ is the number of years which the teacher has received a passing evaluation score and $ Yf $ is the number of years the teacher has received a failing evaluation score. Martha has been teaching for 24 years. She has had passing scores during 17 of those years, neutral scores during 4, and failing scores during 3. Based on the formula, what should her salary be?

To solve this you need to extract the values $ Yp $ and $ Yf $ from the paragraph, then insert them into the given formula and solve. From the text we can see that $ Yp=17 $ and $ Yf=3 $. Plugging these values into the right hand side we get:

$\begin{align*} \text{Salary} & = 30,000+1,000\cdot17-500\cdot3 \\ & = 30,000+17,000-1,500 \\ & = \$45,500 \end{align*}$

We will work more on setting up word problems in the next lesson.

Practice questions

Simplify the following expressions for the given values

$ \frac{x^{3}-x^{2}}{x^{2}-x}\quad x=3 $
$ \frac{a}{b}+\frac{b}{a}\quad a=3\quad b=4 $
$ x^{2}+y^{2}+x-y\quad x=-1\quad y=3 $

Substitute the given values into the equations to solve for x

$ x=\frac{2\cdot z-y}{y^{2}}\quad y=-2\quad z=-7 $
$ \begin{align*} & x=120\cdot(a-b)+10\cdot (a\cdot b) \\ & a=6 \quad b=2 \end{align*} $

Solve for the given variable by substitution

A company pays annual bonuses based on the following formula: $$ \text{Bonus}=20\cdot t+25\cdot p+ 50\cdot y $$ where t is the number of overtime hours an employee worked, p is the productivity score (between 1 and 100) the employee received, and y is the number of years the employee has been with the company. Jenji has worked for the company for 8 years. She received a productivity score of 81, and worked 112 hours of overtime. What bonus will Jenji receive?
Sam works for the same company. He has worked there for 3 years. He received a productivity score of 93 and logged 45 hours of overtime. What bonus will Sam receive?
Every member of a sports team completes three drills and they are given scores between 1 and 100 based on their success for each drill. They are also timed running the 40-yd dash. Their overall score is the average of the three drill scores divided by their 40-yd dash time. Seth scores 67 on the first drill, 95 on the second drill, and 81 on the third drill and runs a 4.80 second forty-yard dash. What overall score does Seth earn?

Solutions

$ \frac{3^{3}-3^{2}}{3^{2}-3}=\frac{27-9}{9-3}=\frac{18}{6}=3 $
$ \frac{3}{4}+\frac{4}{3}=\frac{9}{12}+\frac{16}{12}=\frac{25}{12} $
Solution: $$ \begin{align*} & (-1)^{2}+3^{2}+(-1)-3= \\ & 1+9-1-3=6 \end{align*} $$
Solution: $$ \begin{align*} x & = \frac{2\cdot(-7)-(-2)}{(-2)^{2}} \\[5pt] & = \frac{-14+2}{4} \\[5pt] & = \frac{-12}{4} \\[5pt] & = -3 \end{align*} $$
Solution: $$ \begin{align*} x & = 120(6-2)+10(6\cdot2) \\ & = 120(4)+10(12) \\ & = 480+120=600 \end{align*} $$
$ 20(112)+25(81)+50(8)=\$4,665 $
$ 20(45)+25(93)+50(3)=\$3,375 $
Solution: $$ \begin{align*} (\frac{67+95+81}{3})/4.8 &= (\frac{243}{3})/4.8 \\ &= 81/4.8=16.875 \end{align*} $$

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Substituting values

Expressions vs equations

Substituting values into expressions

Practice questions

Solutions

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