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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.
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.
What's the value of 4x + 5 when x = 3?
Let's break it down: 4x + 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 = 17
Long story short: to substitute a value for a variable, replace every copy of the variable with the value enclosed in parentheses.
What's the value of 2y + y2 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?
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.
Consider the formula C = 5xy. 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, 5xy, for x = 2 and y = 3.
C = 5(2)(3) = 30
We 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.
Find the area of a square with sides that are 4 cm long.
The area of a square is given by the formula A = s2, 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 cm2
We will now let r represent rock star, which is what we are for finding the area of this square.