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Study Guide

Why did we create this toolbox of ways to rearrange expressions into equivalent expressions? Seems pointless, doesn't it? Rest assured, there *is* a point. Here it is:

**If you have two expressions that are equivalent, you can substitute one for another in an expression, equation, or formula.**

What good does *that* do us? Ooh, aren't you about to find out.

### Evaluating Expressions by Substitution

Remember that a variable is like an empty box that's waiting for a number. Have you ever seen a box wait? Those things have unbelievable staying power.We call it

**substitution**when we put a number into the box.### Sample Problem

What's the value of 4

*x*+ 5 when*x*= 3?Let's break it down: 4

*x*+ 5 is the same as 4 · ☐ + 5, so write 3 in the box:4(3) + 5

After substituting values for the variables in an expression, we can

**evaluate**the expression by working out the arithmetic.4(3) + 5 =

12 + 5 = 17Long story short: to substitute a value for a variable, replace every copy of the variable with the value enclosed in parentheses.

### Sample Problem

What's the value of 2

*y*+*y*^{2}when*y*= 4?Here we go: replace every occurrence of

*y*with 4:2(4) + (4)

^{2}4ou see? What did 4ou say? We can stop now? Oh. Thank 4ou.

2(4) + (4)

^{2}=

8 + 16 = 24**Be Careful:**Make sure to put parentheses around values when substituting for variables. There can be some mix-ups with negative signs otherwise. We don't want no more mix-ups. Not after that failed bank heist. You hear that, Ira?### Evaluating Formulas by Substitution

As we mentioned, a formula has a dependent variable on one side and an expression involving the independent variable(s) on the other side. To evaluate a formula, we evaluate the expression containing the independent variable(s). The result is the value of the dependent variable. Ready, Miss Independent? Let's make you Miss Dependent.### Sample Problem

Consider the formula

*C*= 5*xy*. Find*C*if*x*= 2 and*y*= 3.Since

*C*is the dependent variable, that'll be our result. Now, how do we find its value? We evaluate the expression on the other side of the equal sign, 5*xy*, for*x*= 2 and*y*= 3.*C*= 5(2)(3) = 30We now know

*C*'s value is 30. However, we don't know if that's in dollars or pesos—no getting excited just yet.We can use formulas to answer all sorts of questions. Like, "Why is the sky blue?" or "What makes birds sing?" or "What happened to Julia Stiles' career?" More practically, they can help us solve something like the problem below.

### Sample Problem

Find the area of a square with sides that are 4 cm long.

The area of a square is given by the formula

*A*=*s*^{2}, where*A*is the area of the square and*s*is the length of a side. In this case, the length of a side is 4, and we substitute 4 for*s*and evaluate the area formula:*A*= (4)^{2}= 16 cm^{2}We will now let

*r*represent rock star, which is what we are for finding the area of this square.

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