Lesson 4

Equations and Their Solutions

  • Let’s recall what we know about solutions to equations.

4.1: What is a Solution?

A granola bite contains 27 calories. Most of the calories come from \(c\) grams of carbohydrates. The rest come from other ingredients. One gram of carbohydrate contains 4 calories.

The equation \(4c + 5= 27\) represents the relationship between these quantities.

  1. What could the 5 represent in this situation?
  2. Priya said that neither 8 nor 3 could be the solution to the equation. Explain why she is correct.
  3. Find the solution to the equation.

4.2: Weekend Earnings

Jada has time on the weekends to earn some money. A local bookstore is looking for someone to help sort books and will pay \$12.20 an hour. To get to and from the bookstore on a work day, however, Jada would have to spend \$7.15 on bus fare.  

  1. Write an equation that represents Jada’s take-home earnings in dollars, \(E\), if she works at the bookstore for \(h\) hours in one day.
  2. One day, Jada takes home \$90.45 after working \(h\) hours and after paying the bus fare. Write an equation to represent this situation.
  3. Is 4 a solution to the last equation you wrote? What about 7?
    • If so, be prepared to explain how you know one or both of them are solutions.
    • If not, be prepared to explain why they are not solutions. Then, find the solution.
  4. In this situation, what does the solution to the equation tell us?

Jada has a second option to earn money—she could help some neighbors with errands and computer work for \$11 an hour. After reconsidering her schedule, Jada realizes that she has about 9 hours available to work one day of the weekend.

Which option should she choose—sorting books at the bookstore or helping her neighbors? Explain your reasoning.

4.3: Calories from Protein and Fat

One gram of protein contains 4 calories. One gram of fat contains 9 calories. A snack has 60 calories from \(p\) grams of protein and \(f\) grams of fat.

The equation \(4p+9f = 60\) represents the relationship between these quantities. 

  1. Determine if each pair of values could be the number of grams of protein and fat in the snack. Be prepared to explain your reasoning.

    1. 5 grams of protein and 2 grams of fat
    2. 10.5 grams of protein and 2 grams of fat
    3. 8 grams of protein and 4 grams of fat
  2. If there are 6 grams of fat in the snack, how many grams of protein are there? Show your reasoning.
  3. In this situation, what does a solution to the equation \(4p+9f = 60\) tell us? Give an example of a solution.


An equation that contains only one unknown quantity or one quantity that can vary is called an equation in one variable

For example, the equation \(2\ell + 2w = 72\) represents the relationship between the length, \(\ell\), and the width, \(w\), of a rectangle that has a perimeter of 72 units. If we know that the length is 15 units, we can rewrite the equation as:

\(2(15) + 2w = 72\).

This is an equation in one variable, because \(w\) is the only quantity that we don't know. To solve this equation means to find a value of \(w\) that makes the equation true.

In this case, 21 is the solution because substituting 21 for \(w\) in the equation results in a true statement. 

\(\begin {align}2(15) + 2w &=72\\ 2(15)+2(21) &= 72\\ 30 + 42 &=72\\ 72&=72 \end{align}\)

An equation that contains two unknown quantities or two quantities that vary is called an equation in two variables. A solution to such an equation is a pair of numbers that makes the equation true. 

Suppose Tyler spends \$45 on T-shirts and socks. A T-shirt costs \$10 and a pair of socks costs \$2.50. If \(t\) represents the number of T-shirts and \(p\) represents the number of pairs of socks that Tyler buys, we can can represent this situation with the equation:

\(10t + 2.50p = 45\)

This is an equation in two variables. More than one pair of values for \(t\) and \(p\) make the equation true.

\(t=3\) and \(p=6\) 

\(\begin {align} 10(3) + 2.50(6) &= 45\\ 30 + 15 &=45\\ 45&=45 \end{align}\)

\(t=4\) and \(p=2\) 

\(\begin {align} 10(4) + 2.50(2) &= 45\\ 40 + 5 &=45\\ 45&=45 \end{align}\)

\(t=2\) and \(p=10\) 

\(\begin {align} 10(2) + 2.50(10) &= 45\\ 20 + 25 &=45\\ 45&=45 \end{align}\)

In this situation, one constraint is that the combined cost of shirts and socks must equal \$45. Solutions to the equation are pairs of \(t\) and \(p\) values that satisfy this constraint.

Combinations such as \(t=1\) and \(p = 10\) or \(t=2\) and \(p=7\) are not solutions because they don’t meet the constraint. When these pairs of values are substituted into the equation, they result in statements that are false.

Glossary Entries

  • constraint

    A limitation on the possible values of variables in a model, often expressed by an equation or inequality or by specifying that the value must be an integer. For example, distance above the ground \(d\), in meters, might be constrained to be non-negative, expressed by \(d \ge 0\).

  • model

    A mathematical or statistical representation of a problem from science, technology, engineering, work, or everyday life, used to solve problems and make decisions.