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Equations that Describe Patterns

Describe numerical sequences by finding a rule.

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Equations that Describe Patterns

Suppose that you have 45 minutes to do your math homework, and your teacher has assigned 15 problems. To find out, on average, how many minutes you can spend on each problem, what equation could you set up? Also, if you set up an equation and solve it to find the answer, how will you know that your answer is correct?

Writing Algebraic Equations

When an algebraic expression is set equal to another value, variable, or expression, a new mathematical sentence is created. This sentence is called an equation. An algebraic equation is a mathematical sentence connecting an expression to a value, a variable, or another expression with an equal sign (=).

These words can be used to symbolize the equal sign:

Exactly, equivalent, the same as, identical, is

Let's solve the following problems using an algebraic equation:

  1. Suppose there is a concession stand at a theme park selling burgers and French fries. Each burger costs $2.50 and each order of french fries costs $1.75. You and your family will spend exactly $25.00 on food. How many burgers can be purchased? How many orders of fries? How many of each type can be purchased if your family plans to buy a combination of burgers and fries?

The word "exactly" lends a clue to the type of mathematical sentence you will need to write to model this situation. The word "exactly" is synonymous with "equal", so this word is directing us to write an equation.

Let's let \begin{align*}b\end{align*} stand for burgers and \begin{align*}f\end{align*} stand for french fries, the two unknown quantities. Since each burger costs $2.50 and each order of french fries costs $1.75, using what you learned in 1.1 and 1.6, one side of the equation will be:

\begin{align*}2.5b + 1.75f\end{align*} 

Since your family wants to spend exactly $25.00 on food, the expression above must be set equal to 25. Thus we get the equation:

\begin{align*}2.5b + 1.75f=25\end{align*}

If you only want to purchase burgers, you can ignore the part of the equation that relates to fries (\begin{align*}1.75f\end{align*}) and if you want to only purchase fries, you can ignore the part of the equation that relates to burgers (\begin{align*}2.5b\end{align*}). If you want a combination, you have to determine which numbers will satisfy the equation we found above. 

  1. Using the information from problem 1, how many burgers can be purchased with $25.00?

Step 1: Choose a variable to represent the unknown quantity, say \begin{align*}b\end{align*} for burgers.

Step 2: Write an equation to represent the situation: \begin{align*}2.50 b = 25.00\end{align*}.

Step 3: Think. What number multiplied by 2.50 equals 25.00?

That number is 10. Checking an answer to an equation is almost as important as the equation itself. By substituting the value for the variable, you are making sure both sides of the equation balance. Let's check that 10 is the solution to our equation by substituting it back in for \begin{align*}b\end{align*}.

\begin{align*}2.50 (10) = 25.00\end{align*}

\begin{align*}25.00 = 25.00\end{align*}

Since these numbers are equal, 10 is the solution. Your family can purchase exactly ten burgers.

  1. Is \begin{align*}z = 3\end{align*} a solution to \begin{align*}z^2 + 2z = 8\end{align*}?

Begin by substituting the value of 3 for \begin{align*}z\end{align*}.

\begin{align*}3^2 + 2(3) &= 8\\ 9 + 6 &= 8\\ 15 &= 8\end{align*}

Because \begin{align*}15 = 8\end{align*} is NOT a true statement, we can conclude that \begin{align*}z = 3\end{align*} is not a solution to \begin{align*}z^2 + 2z = 8\end{align*}.

  1. Check that \begin{align*}x = 5\end{align*} is the solution to the equation \begin{align*}3x + 2 = -2x + 27\end{align*}.

To check that \begin{align*}x = 5\end{align*} is the solution to the equation, substitute the value of 5 for the variable, \begin{align*}x\end{align*}:

\begin{align*}3x + 2 &= -2x + 27\\ 3 \cdot x + 2 &= -2 \cdot x + 27\\ 3 \cdot 5 + 2 &= -2 \cdot 5 + 27\\ 15 + 2 &= -10 + 27\\ 17 &= 17\end{align*}

Because \begin{align*}17 = 17\end{align*} is a true statement, we can conclude that \begin{align*}x = 5\end{align*} is a solution to \begin{align*}3x + 2 = -2x + 27\end{align*}.


Example 1

Earlier, you were told that you have 45 minutes to complete the 15 math problems your teacher assigned. What equation could you set up to find out on average how many minutes you can spend on each problem? How do you know that the answer is correct?

Let \begin{align*}m\end{align*} be the unknown that represents the number of minutes you are spending on each problem. There are 15 problems to so the total number of time you will take can be represented by:


The equation then will be:


What number multiplied by 15 equals 45? That number is 3. So you will have on average 3 minutes to spend on each problem. 

To check that your answer is correct, substitute in 3 for the variable \begin{align*}m\end{align*}:

\begin{align*}15m=45\\ 15(3)=45\\ 45=45\end{align*}

Since \begin{align*}45=45\end{align*}, the answer is correct.

Example 2

Translate the statement into an algebraic equation: 9 less than twice a number is 33.

Let “a number” be \begin{align*}n\end{align*}. So, twice a number is \begin{align*}2n\end{align*}.

Nine less than that is \begin{align*}2n - 9\end{align*}.

The word "is" means the equal sign, so \begin{align*}2n - 9 = 33\end{align*}.

Example 3

Translate the statement into an algebraic equation: Five more than four times a number is 21.

Let “a number” be \begin{align*}x\end{align*}. So five more than four times a number is 21 can be written as: \begin{align*}4x + 5 = 21.\end{align*}

Example 4

Translate the statement into an algebraic equation: $20.00 was one-quarter of the money spent on pizza.

Let “of the money” be \begin{align*}m\end{align*}. The equation could be written as \begin{align*}\frac{1}{4} m = 20.00.\end{align*}

Example 5

Find out how much money was spent on the pizza in Example 4.

\begin{align*}\frac{1}{4} m = 20.00\end{align*}

Think: One-quarter can also be thought of as divide by four. What divided by 4 equals 20.00?

The solution is 80. So, the money spent on pizza was $80.00.


In 1–3, define the variables and translate the following statements into algebraic equations.

  1. Peter’s Lawn Mowing Service charges $10 per job and $0.20 per square yard. Peter earns $25 for a job.
  2. Renting the ice-skating rink for a birthday party costs $200 plus $4 per person. The rental costs $324 in total.
  3. Renting a car costs $55 per day plus $0.45 per mile. The cost of the rental is $100.

In 4–7, check that the given number is a solution to the corresponding equation.

  1. \begin{align*}a = -3; \ 4a + 3 = -9\end{align*}
  2. \begin{align*}x = \frac{4}{3}; \ \frac{3}{4}x + \frac{1}{2} = \frac{3}{2}\end{align*}
  3. \begin{align*}y = 2; \ 2.5y - 10.0 = -5.0\end{align*}
  4. \begin{align*}z = -5; \ 2(5 - 2z) = 20 - 2(z - 1)\end{align*}

In 8-12, find the value of the variable.

  1. \begin{align*}m + 3 = 10\end{align*}
  2. \begin{align*}6 \times k = 96\end{align*}
  3. \begin{align*}9 - f = 1\end{align*}
  4. \begin{align*}8h = 808\end{align*}
  5. \begin{align*}a + 348 = 0\end{align*}

In 13-15, answer by writing an equation and solving for the variable.

  1. You are having a party and are making sliders. Each person will eat 5 sliders. There will be seven people at your party. How many sliders do you need to make?
  2. The cost of a Ford Focus is 27% of the price of a Lexus GS 450h. If the price of the Ford is $15,000, what is the price of the Lexus?
  3. Suppose your family will purchase only orders of French fries using the information found in the opener of this lesson. How many orders of fries can be purchased for $25.00?

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.8. 

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Algebraic equation An algebraic equation is a mathematical sentence connecting an expression to a value, a variable, or another expression with an equal sign (=).

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