Your summer landscaping job pays a fixed rate of $20 per job plus $4 an hour. How much total money would you would make if it takes you 3 hours to complete a single job?

### Evaluating Algebraic Expressions and Equations

You have probably seen letters in a mathematical expression, such as \begin{align*}3x-8\end{align*}**variables,** represent an unknown number. One of the goals of algebra is to solve various equations for a variable. Typically, \begin{align*}x\end{align*}

To evaluate an expression or equation, we would need to substitute in a given value for the variable and test it. In order for the given value to be true for an equation, the two sides of the equation must simplify to the same number.

Let's evaluate the following expressions and equations.

- Evaluate \begin{align*}2x^2-9\end{align*}
2x2−9 for \begin{align*}x=-3\end{align*}x=−3 .

We know that \begin{align*}2x^2-9\end{align*}

\begin{align*}&2(-3)^2-9 \rightarrow (-3)^2=-3 \cdot -3=9\\
&2(9)-9\\
&18-9\\
&9\end{align*}

You will need to remember that when squaring a negative number, the answer will always be positive. There are three different ways to write multiplication: \begin{align*}2 \times 9, 2 \cdot 9\end{align*}

- Determine if \begin{align*}x=5\end{align*}
x=5 is a solution to \begin{align*}3x-11=14\end{align*}3x−11=14 .

Even though the directions are different, this problem is almost identical to #1 above. However, this is an equation because of the equals sign. Both sides of an equation must be equal to each other in order for it to be true. Plug in 5 everywhere there is an \begin{align*}x\end{align*}

\begin{align*}& \ ?\\
3(5)-11 &= 14\\
15-11 & \neq 14\\
4 &\neq 14\end{align*}

Because \begin{align*}4 \ne 14\end{align*}

- Determine if \begin{align*}t=-2\end{align*}
t=−2 is a solution to \begin{align*}7t^2-9t-10=36\end{align*}7t2−9t−10=36 .

Here, \begin{align*}t\end{align*}

\begin{align*}& \ ?\\
7(-2)^2-9(-2)-10 &= 36\\
& \ ?\\
7(4)+18-10 &=36\\
& \ ?\\
28+18-10 &= 36\\
36 &= 36\end{align*}

-2 is a solution to this equation.

### Examples

#### Example 1

Earlier, you were given a problem about your summer landscaping job.

Rewrite the sentence as an algebraic expression. $20 plus $4 an hour, would be \begin{align*}20+4h\end{align*}, where *h* equals the number of hours you work. Then, evaluate the expression for \begin{align*}h = 3\end{align*}.

\begin{align*}20+4(3)=20+12=32\end{align*}

You will make a total of $32 for this particular job.

#### Example 2

Evaluate \begin{align*}s^3-5s+6\end{align*} for \begin{align*}s=4\end{align*}.

Plug in 4 everywhere there is an \begin{align*}s\end{align*}.

\begin{align*}&4^3-5(4)+6\\ &64-20+6\\ &50\end{align*}

#### Example 3

Determine if \begin{align*}a=-1\end{align*} is a solution to \begin{align*}4a-a^2+11=-2-2a\end{align*}.

Plug in -1 for \begin{align*}a\end{align*} and see if both sides of the equation are the same.

\begin{align*}& \ ? \\ 4(-1)-(-1)^2+11 &= -2-2(-1)\\ & \ ? \\ -4-1+11 &=-2+2\\ 6 & \neq 0\end{align*}

Because the two sides are not equal, -1 is not a solution to this equation.

### Review

Evaluate the following expressions for \begin{align*}x = 5\end{align*}.

- \begin{align*}4x-11\end{align*}
- \begin{align*}x^2+8\end{align*}
- \begin{align*}\frac{1}{2}x+1\end{align*}

Evaluate the following expressions for the given value.

- \begin{align*}-2a+7; a=-1\end{align*}
- \begin{align*}3t^2-4t+5; t=4\end{align*}
- \begin{align*}\frac{2}{3}c-7; c=-9\end{align*}
- \begin{align*}x^2-5x+6; x=3\end{align*}
- \begin{align*}8p^2-3p-15; p=-2\end{align*}
- \begin{align*}m^3-1; m=1\end{align*}

Determine if the given values are solutions to the equations below.

- \begin{align*}x^2-5x+4=0; x=4\end{align*}
- \begin{align*}y^3-7=y+3; x=2\end{align*}
- \begin{align*}7x-3=4; x=1\end{align*}
- \begin{align*}6z+z-5=2z+12;z=-3\end{align*}
- \begin{align*}2b-5b^2+1=b^2;b=6\end{align*}
- \begin{align*}-\frac{1}{4}g+9=g+15;g=-8\end{align*}

Find the value of each expression, given that \begin{align*}a=-1,b=2,c=-4\end{align*}, and \begin{align*}d=0\end{align*}.

- \begin{align*}ab-c\end{align*}
- \begin{align*}b^2+2d\end{align*}
- \begin{align*}c+\frac{1}{2}b-a\end{align*}
- \begin{align*}b(a+c)-d^2\end{align*}

For problems 20-25, use the equation \begin{align*}y^2+y-12=0\end{align*}.

- Is \begin{align*}y = 4\end{align*} a solution to this equation?
- Is \begin{align*}y = -4\end{align*} a solution to this equation?
- Is \begin{align*}y = 3\end{align*} a solution to this equation?
- Is \begin{align*}y = -3\end{align*} a solution to this equation?
- Do you think there are any other solutions to this equation, other than the ones found above?
**Challenge**Using the solutions you found from problems 20-23, find the sum of these solutions and their product. What do you notice?

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 1.4