Writing and Solving Equations from Word Problems

📝 Practice Worksheets

Writing Equations and Solving Write and Solve Equations Real-World Equation Practice

🛠️ Related Tools

Equation Balancer Machine Equation Solving Coach

🔑 Key Concepts

Define a variable to represent the unknown quantity.
Identify the known quantities and how they are related.
Translate totals, equal groups, and fractional parts into algebraic equations.
Use parentheses when a fraction or factor applies to an entire group.
Perform the same operation on both sides to keep an equation balanced.
Interpret the solution using the units and context of the problem.

✏️ Worked Examples — Representing and Solving Word Problems with Equations

Example 1 — Spencer’s Weekly Biking Goal

Spencer biked 6 miles on Tuesday, 8 miles on Thursday, and 11 miles on Saturday. He wants to bike a total of 35 miles for the week. Let \(x\) represent the number of miles Spencer bikes on Sunday. Which equation does not represent the situation?

1. Find the known total

\(6+8+11=25\)

Spencer has already biked 25 miles. The remaining distance is represented by \(x\).

2. Compare the equations

\(6+8+11+x=35\)

\(x+25=35\)

\(35-25=x\)

These three equations all show that the completed miles plus the Sunday miles equal 35.

3. Identify and solve

\(35=6+8+11-x\)

This equation does not represent the situation because it subtracts the Sunday miles from the completed miles.

\(25+x=35\)

\(25+x\ {\color{red}{-25}}=35\ {\color{red}{-25}}\)

\(x=10\)

Spencer must bike 10 miles on Sunday.

Example 2 — Original Price of a Jacket

Liam buys a jacket and a pair of shoes for \(\displaystyle \frac{4}{5}\) of their combined original price. He pays 💲64.00 for both items. The original price of the shoes is 💲30. Let \(j\) represent the original price of the jacket. Write an equation and solve for \(j\).

1. Represent the total

The combined original price is the shoe price plus the jacket price:

\(30+j\)

Liam pays \(\displaystyle \frac{4}{5}\) of that entire amount.

2. Write the equation

\(\displaystyle \frac{4}{5}(30+j)=64\)

The parentheses show that the fraction applies to the combined original price of both items.

3. Solve and interpret

Multiply both sides by \(\displaystyle \frac{5}{4}\):

\(\displaystyle {\color{red}{\frac{5}{4}}} \left[\frac{4}{5}(30+j)\right] = 64{\color{red}{\left(\frac{5}{4}\right)}} \)

\(30+j=80\)

\(30+j\ {\color{red}{-30}}=80\ {\color{red}{-30}}\)

\(j=50\)

The jacket’s original price was 💲50.

Example 3 — Movie Ticket and Snack Costs

Six members of the Ramirez family go to a movie theater. Each person buys one movie ticket and a snack combo that costs 💲4.75. The total cost for the group is 💲93. Let \(x\) represent the price of one movie ticket. Write an equation and solve for \(x\).

1. Represent each cost

Cost of six tickets:

\(6x\)

Cost of six snack combos:

\(6(4.75)\)

\(6(4.75)=28.50\)

2. Write and simplify

\(6x+6(4.75)=93\)

\(6x+28.50=93\)

The ticket cost and snack cost are added because together they make the group’s total.

3. Solve and interpret

\(6x+28.50\ {\color{red}{-28.50}}=93\ {\color{red}{-28.50}}\)

\(6x=64.50\)

\(\displaystyle \frac{6x}{\color{red}{6}}=\frac{64.50}{\color{red}{6}}\)

\(x=10.75\)

One movie ticket costs 💲10.75.

Check: \(6(10.75)+6(4.75)=64.50+28.50=93\).

🧠 Math Vocabulary

Variable: A symbol used to represent an unknown quantity.
Equation: A mathematical statement showing that two expressions have equal values.
Model: A mathematical representation of a real-world situation.
Grouped quantity: A collection of terms treated as one amount, often shown with parentheses.
Fractional coefficient: A fraction that multiplies a variable or expression.
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 translate real-world situations into algebraic equations and solve for unknown quantities. The structure of each equation depends on the relationship described in the problem. Totals may be represented by addition, repeated equal costs by multiplication, and fractional portions by multiplying a grouped expression. After solving, students interpret the answer using the correct real-world units.

📚 What You Should Already Know

Before this lesson, students should understand variables, basic algebraic expressions, and one-step and two-step equations. Students should also be comfortable with decimal operations, fraction multiplication, inverse operations, and identifying totals and equal groups in word problems.

🚀 What Comes Next

Next, students can write and solve more complex equations involving percentages, rates, proportions, distribution, and variables on both sides. These modeling skills also prepare students for inequalities, formulas, and multi-step real-world applications.

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