1.12: Solve Real-World Problems by Writing and Solving Single-Variable Equations
Did you know that you can use an equation to solve a problem? Creating a model for a problem may also include methods such as drawing a diagram or picture or making a table or chart.
Take a look at this dilemma.
The triangles below were constructed using toothpicks. Determine the number of toothpicks needed to construct twenty triangles.
Do you know how to figure this out? Pay attention to this Concept. Then you will know how to solve this dilemma.
Guidance
Sometimes if you think of a problem in terms of words and parts it will be easier to write an equation and solve it. Writing a verbal model is similar to making a plan for solving a problem. When you write a verbal model, you are paraphrasing the information stated in the problem. After writing a verbal model, insert the values from the problem to write an equation. Then, use mental math or an inverse operation to solve it.
Take a look at this situation.
Monica purchased a pair of tennis shoes on sale for $65.99. The shoes were originally $99.00. Use a verbal model to write and solve an equation to determine the amount of money Monica saved by purchasing the shoes on sale
First write a verbal model to represent the problem.
Verbal Model: \begin{align*}\text{Sale Price} + \text{Amount Saved} = \text{Original Price}\end{align*}
Let “\begin{align*}s\end{align*}” represent the amount saved.
Equation: \begin{align*}65.99 + s = 99.00\end{align*}
Solution: Recall that to solve for “\begin{align*}s\end{align*},” complete the inverse operation. Since addition is used in the equation, use subtraction to solve.
It makes sense to subtract 65.99 from 99.00.
\begin{align*}99.00 - 65.99 = \$33.01\end{align*}
This is the answer.
Write an equation for each situation and solve it.
Example A
Mary had $12.00 and she spent some amount. She has $4.50 left over. How much did she spend?
Solution: \begin{align*}12 - x = 4.50, x = 7.50\end{align*}
Example B
John spent twice as much as Mary did. How much did he spend?
Solution: \begin{align*}2(7.50) = $15.00\end{align*}
Example C
A number and sixteen is equal to forty-five.
Solution: \begin{align*}x + 16 = 45, x = 29\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
As you can see, three toothpicks were needed to construct one triangle. Two more were needed to construct the second triangle. Therefore, five toothpicks were used to make two triangles. Continue to make more triangles along the row. Each time you construct a new triangle, record the number of toothpicks used on a chart.
Triangle #: | Toothpick # |
---|---|
1 | 3 |
2 | 5 |
3 | 7 |
4 | 9 |
5 | 11 |
6 | 13 |
7 | 15 |
8 | 17 |
9 | 19 |
10 | 21 |
Looking at the table, you can identify a pattern. You can see that two toothpicks are needed each time a new triangle is constructed. You can write a verbal model to express this amount.
Total Number of Toothpicks Needed = Two Times the Number of Triangles + One Toothpick
Let \begin{align*}n =\end{align*} number of triangles
Total Number of Toothpicks Needed \begin{align*}= 2n + 1\end{align*}
To determine the number of toothpicks needed to construct twenty triangles, substitute twenty for the variable.
\begin{align*}& 2n + 1\\ & 2(20) + 1\\ & 40 + 1\\ & 41\end{align*}
41 toothpicks are needed to construct twenty triangles.
Vocabulary
- Equation
- a group of numbers, operations and variables where the quantity on one side of the equal sign is the same as the quantity on the other side of the equal sign.
- Inverse Operation
- the opposite operation. Equation can often be solved by using an inverse operation.
- Verbal Model
- using words to decipher the mathematical information in a problem. An equation can often be written from a verbal model.
Guided Practice
Here is one for you to try on your own.
The cost to run a thirty second commercial on prime time television is seven hundred fifty-thousand dollars. Use a verbal model to write and solve an equation to determine the cost per second.
Solution
Verbal Model: \begin{align*}\frac{\text{Total Cost}}{\text{Number of Seconds}} = \text{Cost per Second}\end{align*}
Let “\begin{align*}x\end{align*}” represent the unknown cost per second.
Equation: \begin{align*}\frac{\$750,000}{30} = x\end{align*}
Solution: To solve, divide 750,000 by 30.
\begin{align*}\frac{750,000}{30} &= x\\ 25,000 &= x\end{align*}
Now remember that we were talking about money in this problem. So our answer needs to be written as a money amount.
The answer is that is costs $25,000 per second for a thirty second commercial.
Video Review
Practice
Directions: Write an equation for each situation and then solve for the variable. Each problem will have two answer to it.
1. An unknown number and three is equal to twelve.
2. John had a pile of golf balls. He lost nine on the course. If he returned home with fourteen golf balls, how many did he start with?
3. Some number and six is equal to thirty.
4. Jessie owes her brother some money. She earned nine dollars and paid off some of her debt. If she still owes him five dollars, how much did she owe to begin with?
5. A farmer has chickens. Six of them went missing during a snowstorm. If there are twelve chickens left, how many did he begin with before the storm?
6. Gasoline costs four dollars per gallon. Kerry put many gallons in his car over a long car trip. If he spent a total of $140.00 on gasoline, how many gallons did he need for the trip?
7. Twenty-seven times a number is 162. What is the number?
8. Marsha divided cookies into groups of 12. If she had 6 dozen cookies when she was done, how many cookies did she start with?
9. The coach divided the students into five teams. There were fourteen students on each team. How many students did the coach begin with?
10. A number plus nineteen is equal to forty.
Algebraic Equation
An algebraic equation contains numbers, variables, operations, and an equals sign.Consecutive
The term consecutive means "one after another." An example of consecutive numbers is 1, 2, and 3. An example of consecutive even numbers would be 2, 4, and 6. An example of consecutive odd numbers would be 1, 3, and 5.Equation
An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.Integer
The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...Inverse Operation
Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.Verbal Model
A verbal expression (or verbal model) uses words to decipher the mathematical information in a problem. An equation can often be written from a verbal model.Image Attributions
Here you'll solve real-world problems by writing and solving single-variable equations.
Concept Nodes:
Algebraic Equation
An algebraic equation contains numbers, variables, operations, and an equals sign.Consecutive
The term consecutive means "one after another." An example of consecutive numbers is 1, 2, and 3. An example of consecutive even numbers would be 2, 4, and 6. An example of consecutive odd numbers would be 1, 3, and 5.Equation
An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.Integer
The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...Inverse Operation
Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.Verbal Model
A verbal expression (or verbal model) uses words to decipher the mathematical information in a problem. An equation can often be written from a verbal model.