Solving Linear Word Problems

📝 Practice Worksheets

Writing and Solving Equations — Practice 1 Writing and Solving Equations — Practice 2 Writing and Solving Equations — Practice 3

🛠️ Related Tools

Equation Balancer Machine Equation Solving Coach

🔑 Key Concepts

Define a variable to represent the unknown quantity.
Translate verbal relationships into algebraic expressions.
Use multiplication to represent equal groups, rates, and repeated quantities.
Combine like terms before isolating the variable.
Use inverse operations while performing the same operation on both sides.
Interpret the solution in the context of the original problem.

✏️ Worked Examples — Write and Solve Equations from Word Problems

Example 1 — Cost per Pound

Garrett paid 💲3.99 for a watermelon and also purchased 3 pounds of cherries. The total cost was 💲9.93. How much did the cherries cost per pound?

Define the variable

Let \(c\) represent the cost of one pound of cherries.

Write the equation

\(3.99+3c=9.93\)

The expression \(3c\) represents 3 pounds of cherries at \(c\) dollars per pound.

Solve and interpret

\(3.99+3c\ {\color{red}{-3.99}}=9.93\ {\color{red}{-3.99}}\)

\(3c=5.94\)

\(\displaystyle \frac{3c}{\color{red}{3}}=\frac{5.94}{\color{red}{3}}\)

\(c=1.98\)

The cherries cost 💲1.98 per pound.

Example 2 — Translating a Verbal Expression

The sum of three times a number and 2 less than four times that same number is 61. Write an equation and determine the value of the number.

Translate the phrases

Three times a number:

\(3n\)

Two less than four times the number:

\(4n-2\)

Write and simplify

\(3n+(4n-2)=61\)

\(7n-2=61\)

The variable terms \(3n\) and \(4n\) combine to make \(7n\).

Solve

\(7n-2\ {\color{red}{+2}}=61\ {\color{red}{+2}}\)

\(7n=63\)

\(\displaystyle \frac{7n}{\color{red}{7}}=\frac{63}{\color{red}{7}}\)

\(n=9\)

The unknown number is \(9\).

Example 3 — Ages of Three Sisters

Jana has an older sister and a younger sister. Her older sister is one year more than twice Jana’s age. Her younger sister is three years younger than Jana. The sum of their three ages is 26. Find Jana’s age.

Represent each age

Let \(a\) represent Jana’s age.

\(\text{Jana}=a\)

\(\text{Older sister}=2a+1\)

\(\text{Younger sister}=a-3\)

Write and simplify

\((2a+1)+(a-3)+a=26\)

\(4a-2=26\)

The three age expressions are added because their total is 26.

Solve and interpret

\(4a-2\ {\color{red}{+2}}=26\ {\color{red}{+2}}\)

\(4a=28\)

\(\displaystyle \frac{4a}{\color{red}{4}}=\frac{28}{\color{red}{4}}\)

\(a=7\)

Jana is 7 years old.

Jana is \(7\), her older sister is \(2(7)+1=15\), and her younger sister is \(7-3=4\). Their ages total \(7+15+4=26\).

🧠 Math Vocabulary

Variable: A symbol used to represent an unknown quantity.
Algebraic expression: A combination of numbers, variables, and operations without an equal sign.
Equation: A mathematical statement showing that two expressions are equal.
Coefficient: A number that multiplies a variable.
Like terms: Terms containing the same variable raised to the same power.
Inverse operations: Operations that undo one another and help isolate a variable.
Interpret: To explain what a mathematical solution means in the context of a problem.

💡 Main Idea

In this lesson, students define a variable, translate verbal relationships into algebraic expressions, write an equation, and solve for an unknown quantity. After solving, the answer must be interpreted in the context of the original situation. This process allows equations to represent costs, numerical relationships, ages, and many other real-world situations.

📚 What You Should Already Know

Students should understand how to solve one-step and two-step equations, combine like terms, and use inverse operations. They should also recognize common verbal phrases such as sum, difference, times, less than, more than, and total.

🚀 What Comes Next

Students will continue writing and solving equations in more complex contexts involving percentages, geometry, rational numbers, and multi-step relationships. These skills also prepare students to write inequalities, rearrange formulas, and create algebraic models.

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