Lesson 1: Representing Situations of the Form px+q=r and p(x+q)=r

Property

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations to solve problems by reasoning about the quantities. This involves assigning a variable to an unknown quantity and translating the problem's context into a mathematical equation.

Examples

• A phone costs 50 dollars more than three times the cost of its case. If the phone costs 350 dollars, what is the cost of the case? Let $c$ be the case's cost. The equation is $3c + 50 = 350$, so $3c = 300$, and $c = 100$. The case costs 100 dollars.
• Sarah has twice as many books as John. Together they have 36 books. How many books does John have? Let $j$ be John's books. The equation is $j + 2j = 36$, so $3j = 36$. John has $j=12$ books.
• A rectangular garden's length is 4 feet longer than its width. If the perimeter is 40 feet, what is the width? Let $w$ be the width. The perimeter is $2w + 2(w+4) = 40$. This simplifies to $4w + 8 = 40$, so $4w = 32$, and $w = 8$ feet.

Explanation

This skill turns a word problem into a solvable math puzzle. You choose a letter to represent the mystery number, then use the clues in the story to build an equation. Solving the equation reveals the answer to the real-world problem.

Property

A relationship of the form $px+q=r$ describes a situation where a total amount, $r$, is the sum of a variable amount, $px$, and a fixed starting amount, $q$. The variable amount is found by multiplying a rate, $p$, by a quantity, $x$.