1.6 Modeling with Equations

1.6.1 Learning Objectives

By the end of this section, you should be able to:

translate verbal information into algebraic expressions and equations;
construct and solve equation models;
organize information using diagrams or tables; and
interpret and evaluate solutions in context.

1.6.2 Building a Mathematical Model

An equation is a mathematical model when it represents a relationship from a practical situation. A reliable modeling process has four stages:

Identify the variable. State what the unknown represents and include units.
Express other unknown quantities. Write each relevant quantity in terms of the chosen variable. A table or diagram may help.
Build the equation. Use the key relationship connecting the quantities.
Solve and interpret. Solve the equation, check the result, and answer the original question in a complete sentence.

1.6.3 Cost and Integer Models

Common modeling relationships include:

\[ \text{cost}=(\text{fixed cost})+(\text{rate})(\text{quantity}), \]

Example 1.28 Fixed and Variable Costs

A school club pays a setup fee of $\$45$ plus $\$3.50$ per printed shirt. The bill is $\$185$. How many shirts were printed?

Let $x$ be the number of shirts:

\[ 45+3.50x=185. \]

\[ 3.50x=140 \quad\Longrightarrow\quad x=40. \]

The club printed $\boxed{40\text{ shirts}}$.

Exercise 1.34 Translate: a streaming service charges $\$8$ plus $\$2$ per rental, for a total of $\$24$.

Exercise 1.35 Three consecutive integers have sum $72$. Find them.

1.6.4 Interest and Motion Models

Simple interest can be modeled by

\[ \text{simple interest}=(\text{principal})(\text{rate})(\text{time}). \]

Motion problems use the relationship

\[ \text{distance}=(\text{rate})(\text{time}). \]

Example 1.29 Distance, Rate, and Time

Two cyclists start $84$ km apart and ride toward each other. Their speeds are $18$ km/h and $24$ km/h. When do they meet?

Let $t$ be the travel time:

\[ 18t+24t=84. \]

\[ 42t=84 \quad\Longrightarrow\quad \boxed{t=2\text{ hours}}. \]

Exercise 1.36 A vehicle travels $210$ km in $3$ hours. Find its average speed.

1.6.5 Work-Rate Models

For a task completed at a constant rate,

\[ \text{work rate}=\frac{1}{\text{time for one complete job}}. \]

Example 1.30 Working Together

One printer completes a batch in $6$ minutes and another in $10$ minutes. How long do they take together?

Let $t$ be the time together:

\[ \frac16+\frac1{10}=\frac1t. \]

\[ \frac4{15}=\frac1t \quad\Longrightarrow\quad \boxed{t=\frac{15}{4}=3.75\text{ minutes}}. \]

Exercise 1.37 A machine completes a task in $5$ hours. What fraction of the task does it complete per hour?

1.6.6 Mixture Models

Mixture problems use the relationship

\[ \text{amount of ingredient} =(\text{concentration})(\text{total volume}). \]

Example 1.31 Mixture

How much pure juice should be added to $20$ L of a $30\%$ juice drink to make a $50\%$ mixture?

Let $x$ be the litres of pure juice added:

\[ 0.30(20)+x=0.50(20+x). \]

\[ 6+x=10+0.5x \quad\Longrightarrow\quad x=8. \]

Add $\boxed{8\text{ L}}$ of pure juice.

1.6.7 Geometry Models

Geometry models often use perimeter, area, volume, the Pythagorean theorem, or similar triangles. Answers must respect the context. A negative length, for example, is rejected even if it satisfies the algebraic equation.

Example 1.32 Geometry

A rectangle is $7$ cm longer than it is wide and has area $144\text{ cm}^2$. Find its dimensions.

Let $w$ be the width. Then the length is $w+7$:

\[ w(w+7)=144. \]

\[ w^2+7w-144=(w+16)(w-9)=0. \]

The positive solution is $w=9$. The dimensions are

\[ \boxed{9\text{ cm by }16\text{ cm}}. \]

Exercise 1.38 A rectangle is $4$ m longer than it is wide and has area $96\text{ m}^2$. Find its dimensions.

1.6.8 Conceptual Takeaways

Good modeling begins by clearly defining the unknown.
Every expression in an equation should have compatible units and meaning.
Tables and diagrams help reveal relationships.
The algebraic solution must be checked against the original situation.
A mathematically valid root may be impossible in context.
The final response should interpret the answer, not merely state a number.

1.6.9 Skills You Should Be Able to Do

Define variables with units.
Translate words into algebraic expressions and equations.
Model cost, interest, geometry, mixture, work, and motion situations.
Choose and use relevant formulas.
Reject unreasonable or inadmissible solutions.
Check and communicate conclusions in context.

1.6.10 Practice Problems with Solutions

A museum charges a group fee of $\$60$ plus $\$12$ per student. The total is $\$300$. How many students attend?

Show Solution

\[ 60+12x=300 \Longrightarrow 12x=240 \Longrightarrow x=20. \] There are $\boxed{20\text{ students}}$.
Find three consecutive odd integers whose sum is $123$.

Show Solution

Let the integers be $n,n+2,n+4$: \[ n+(n+2)+(n+4)=123. \] \[ 3n+6=123 \Longrightarrow n=39. \] The integers are $\boxed{39,41,43}$.
A rectangular garden is $5$ m longer than it is wide and has area $84\text{ m}^2$. Find its dimensions.

Show Solution

Let $w$ be the width: \[ w(w+5)=84 \Longrightarrow w^2+5w-84=0. \] \[ (w+12)(w-7)=0. \] Rejecting the negative root gives dimensions \[ \boxed{7\text{ m by }12\text{ m}}. \]
A $1.6$-m student casts a $2$-m shadow while a tree casts a $15$-m shadow. Find the tree’s height.

Show Solution

Similar triangles give \[ \frac{h}{15}=\frac{1.6}{2}. \] Thus \[ h=15(0.8)=\boxed{12\text{ m}}. \]
How many litres of a $70\%$ solution should be mixed with $12$ L of a $20\%$ solution to obtain a $40\%$ solution?

Show Solution

Let $x$ be the litres of $70\%$ solution: \[ 0.70x+0.20(12)=0.40(x+12). \] \[ 0.30x=2.4 \Longrightarrow \boxed{x=8\text{ L}}. \]
One scanner processes a collection in $8$ hours and another in $12$ hours. How long do they take together?

Show Solution

\[ \frac18+\frac1{12}=\frac1t \Longrightarrow \frac5{24}=\frac1t. \] Therefore \[ \boxed{t=\frac{24}{5}=4.8\text{ hours}}. \]
A bus travels $300$ km. On the return trip its average speed is $15$ km/h faster, reducing travel time by $1$ hour. Find the outbound speed.

Show Solution

Let $v$ be the outbound speed: \[ \frac{300}{v}-\frac{300}{v+15}=1. \] Multiplying by $v(v+15)$ gives \[ 300(v+15)-300v=v(v+15). \] \[ v^2+15v-4500=0 =(v+75)(v-60). \] The positive solution is $\boxed{60\text{ km/h}}$.
A square display has a uniform border $3$ cm wide. The total area, including the border, is $324\text{ cm}^2$. Find the side length of the inner square.

Show Solution

Let $x$ be the inner side length. The total side length is $x+6$: \[ (x+6)^2=324. \] Since a length is positive, $x+6=18$, so \[ \boxed{x=12\text{ cm}}. \]