Translate between verbal and algebraic expressions and
Apply properties of equality to solve algebraic expressions

Suppose I placed an order for supplies I needed for my classroom. I ordered three boxes of supplies. Each box contained 20 pencils and 3 erasers. I decided to keep two pencils and one eraser for myself. How many of each item is left for the class? By re-writing this verbal description into an algebraic expression, it becomes a bit easier to solve. Recall that expressions are mathematical statements made up of a collection of numbers, variables, and operators. We should identify our numbers, variables and operators necessary for this expression. Since this problem asks for the number of pencils and erasers left for the class, this is a pretty good indicator of our variables. I'll use the letter x to represent one pencil and y to represent one eraser. Variables are letters used to represent an unknown quantity. Each box contains 20 pencils and 3 erasers, or 20x+3y. Since I ordered three boxes, then the expression should look like this, [3(20x+3y)]. And finally, because I kept two pencils and three erasers for myself, I should subtract that from the total, [3(20x+3y)-2x-3y]. I find it most helpful to start with the question, and work backward through the problem to create the expression. Let's try another.

One side of a barn is 54 feet long, and will act as one side of a rectangular pen for the animals in the barn. The farmer would like to make the length of the pen, three times as long as the width of the barn. What are the dimensions of this pen and what length of fence is needed to build this pen? If we start with the question in mind, we could identify variables. Since a rectangular pen has a length and width measurement, I'll use w for width, and l for length. Therefore, the amount of fence needed for the pen would be w+2l because one side of the barn will act as a side of the rectangular pen. Since the farmer would like the length of this pen to be three times the width, we could say w+2(3w) represents the length of fence needed. So, 7w is the total length of fence needed, or 7*54=378 feet is necessary.

Now suppose this farmer already had 500 feet of fence ready for this project, how long could the pen be? Using the same expression as before, we could make an equation, [w+2l=500], and allow 54 to substitute for w, [54+2l=500]. Since we are starting with a pair of expressions that are equal to each other, we could perform mathematical operations simultaneously to both sides of the equal sign, and retain equality of the expressions. Start by subtracting 54 from both sides of the equation. The results are still equivalent, [2l=446]. Then we can divide both sides of the equation by 2, and again the results are equivalent, [l=223]. The length of the pen could be 223 feet long.

There are many properties of equality that can apply to equations that makes it easy to solve for an unknown variable. Essentially, the properties of equality provide a series of inverse operations that allow us to "undo" an equation. With the additive and multiplicative inverses, which amounts to subtraction and division, one can solve for unknown variables. First, clear grouping symbols with distribution, then combine like terms, and finally use additive and multiplicative inverses to isolate the variable.

Let's try some examples not unlike the barn problem from before. This time, we'll start with the equation, rather than a word problem. Suppose 5(x-1)-4=2x-(3-x). Start by clearing the parenthesis by using distribution, [5x-5-4=2x-3+x], then combine like terms, [5x-9=3x-3]. Then, by using additive inverses we can move the terms with variables to one side of the equals sign and move the numbers themselves to the other side. Subtract 3x from both sides, then add 9 to both sides, [2x=6]. Finally, the multiplicative inverse, division, can be used to solve for x, [x=3]. We could check our work by evaluating the original expressions using the order of operations, both sides should be the same, [5(3-1)-4=6 and 2*3-(3-3)=6].

To finish this video, I'll provide a few examples with solutions that you should try on your own. Pause the video after the example appears, and resume playback to check your work.

Examples: \(\mathsf{ 2(x-1) = -2, 3x - 2(x-3) = 15, 2y - (y+10) = 5(y+6) }\)