2.4 Applications of Linear Equations

Scott McDaniel

2.4 Applications of Linear Equations

Learning Objectives

After completing this section, you will be able to:

Translate real-world situations into algebraic equations.
Define variables and write equations that represent relationships between quantities.
Solve one-variable equations in applied contexts.
Interpret and check your solutions in words and units.

Launch – From Words to Equations

Up to now, we have solved many symbolic equations—but real-life problems are often written in words.

To solve applied problems, we must:

Define a variable to represent the unknown quantity.
Translate the relationships into an algebraic equation.
Solve the equation using the properties of equality.
Answer in a complete sentence, using correct units.

Tip: Words like “is,” “equals,” “gives,” or “has” often indicate an equals sign (=).

Consecutive Integer and Page Problems

Consecutive numbers follow one after another.

Consecutive integers: $x,\ x+1,\ x+2$
Consecutive even integers: $x,\ x+2,\ x+4$
Consecutive odd integers: $x,\ x+2,\ x+4$

Example 2.4.1

The sum of three consecutive even integers is 138. Find the integers.

Solution

Let the first even integer be $x$ . The next two are $x+2$ and $x+4$ .

Equation: $x+(x+2)+(x+4)=138$

Combine like terms: $3x+6=138$

Subtract 6: $3x=132$

Divide by 3: $x=44$

✅ Integers: 44, 46, 48

Example 2.4.2

The sum of three consecutive odd integers is 81. Find the integers.

Solution

Let the first odd integer be $x$ . The next two are $x+2$ and $x+4$ .

Equation: $x+(x+2)+(x+4)=81$

$3x+6=81$

$3x=75$

$x=25$

✅ Integers: 25, 27, 29

Example 2.4.3

Two pages that face each other have a sum of 187. What are the page numbers?

Solution

Let the first page be $x$ . The facing page is $x+1$ .

Equation: $x+(x+1)=187$

$2x+1=187$

$2x=186$

$x=93$

✅ Pages: 93 and 94

Try It

Geometry Applications

Example 2.4.4

The second angle is 20° less than the first. The third angle is twice the first. Find all three angles.

Solution

Let the first angle be $x$ .

Second: $x-20$

Third: $2x$

Draw a triangle and label. It does not need to be to scale.

Triangle labeled x, x-20, and 2x (not drawn to scale) — **Figure 2.4A**

Triangle sum: $x+(x-20)+2x=180$

$4x-20=180$

$4x=200$

$x=50$

✅ Angles: 50°, 30°, 100°

Example 2.4.5

A rectangle has a perimeter of 48 cm. The length is 4 cm more than twice the width. Find the length and width.

Solution

Let the width be $w$ . Then the length is $2w+4$ .

Draw and label a rectangle:

Rectangle with labels w and 2w +4 — **Figure 2.4B**

Perimeter formula: $48=2(2w+4)+2w$

$48=6w+8$

$40=6w$

$w=\frac{20}{3}\approx6.67$

Length: $2w+4 = 2\left(\frac{20}{3}\right)+4 = \frac{52}{3}\approx17.33$

✅ Width ≈ 6.67 cm, Length ≈ 17.33 cm

Try It

Comparison and Number Relationships

Many word problems compare quantities using phrases like “less than,” “more than,” and “times.” These phrases translate directly into algebraic operations.

Example phrase: “Five less than three times a number is equal to seven more than twice the number.”

“Five less than” → subtract 5
“Three times a number” → $3x$
“Seven more than twice the number” → $2x+7$

Example 2.4.6

Five less than three times a number equals seven more than twice the number. Find the number.

Solution

Let the number be $x$ .

Equation: $3x-5=2x+7$

Solve:

$x-5=7$

$x=12$

✅ Number: 12

Example 2.4.7

Twice the greater of two consecutive odd integers is 13 more than the lesser. Find the numbers.

Solution

Let the smaller integer be $x$ . Then the greater is $x+2$ .

Equation: $2(x+2)=x+13$

$2x+4=x+13$

$x+4=13$

$x=9$

✅ Integers: 9 and 11

Try It

Totals and Combined Quantities

Many real-world applications involve two parts that add to a total, such as people, distances, money, or objects.

Example 2.4.8

Together, two siblings collected 84 seashells. The older collected 10 more than the younger. Find how many seashells each collected.

Solution

Let the younger sibling be $x$ . Then the older collected $x+10$ .

Equation: $x+(x+10)=84$

$2x+10=84$

$2x=74$

$x=37$

✅ Younger: 37 Older: 47

Example 2.4.9

Amazon and Walmart together employ about 2.3 million people. Walmart employs 700,000 more people than Amazon. How many people do they each employ?

Solution

Let Amazon employ $x$ people. Then Walmart employs $x+700{,}000$ .

Equation: $x+(x+700{,}000)=2{,}300{,}000$

$2x+700{,}000=2{,}300{,}000$

$2x=1{,}600{,}000$

$x=800{,}000$

✅ Amazon: 800,000 Walmart: 1,500,000

Example 2.4.10

In an election, one candidate received twice as many votes as their opponent. Together, they received 9,000 votes. How many votes did each get?

Solution

Let the opponent receive $x$ votes. Then the other candidate received $2x$ votes.

Equation: $x+2x=9{,}000$

$3x=9{,}000$

$x=3{,}000$

✅ Opponent: 3,000 Candidate: 6,000

Try It

Common Mistakes

🚩 Forgetting to define the variable before writing the equation

🚩 Misreading “less than” or “more than” (order matters)

🚩 Forgetting to include all parts of a total

🚩 Leaving answers without units

🚩 Not checking whether the solution makes sense in context

Try It

Key Takeaways

Always define your variable clearly.
Translate words like “sum,” “difference,” “product,” and “quotient” carefully.
Use parentheses to preserve order and meaning.
Check that your solution fits the original situation.
Write answers in complete sentences with appropriate units or labels.

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2.4 Applications of Linear Equations

Launch – From Words to Equations

Consecutive Integer and Page Problems

Geometry Applications

Comparison and Number Relationships

Totals and Combined Quantities

Other Types of Measurement Problems

Common Mistakes

License

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