Real-World Problems with Systems

Lesson 3 of 3: Systems of Linear Equations

In this lesson:

• Turn a word problem into two equations
• Solve, interpret, and check the answer

Learning Objectives for This Lesson

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

1. Translate a real-world scenario into two equations
2. Solve systems arising from real-world problems
3. Interpret the solution in context
4. Decide whether two lines through point pairs intersect
5. Verify a solution algebraically and in context

How Many of Each Ticket?

At a fundraiser, 12 tickets sold for $81. Adult tickets cost $8, child tickets $5.

How many of each were sold?

List the Facts in the Story

Underline what the problem tells you:

• Fact 1: 12 tickets in total
• Fact 2: $81 collected in total

Two unknowns, so we need two facts.

Each Fact Becomes One Equation

Let = adult tickets, = child tickets.

The total count is one equation; the total cost is the other.

Solve the System and Interpret

Substitute into :

• , so , , then

7 adult tickets and 5 child tickets.

The Six-Step Modeling Process Cycle

The algebra is just the middle step.

A Geometry Problem with Two Facts

A rectangle's perimeter is 46 cm. Its length is 5 cm more than twice the width.

• and

Quick Check: Set Up Only

A pair of numbers has a sum of 15 and a difference of 3.

Write two equations. Do not solve yet.

Recognizing Four Common Problem Types

Each type suggests where the two equations come from.

A Mixture Problem with Fractions

Make 8 lb of trail mix at $6/lb from almonds ($10) and raisins ($4).

• and
• Eliminate: lb, lb

Do These Two Lines Cross?

Line 1 through and ; Line 2 through and .

Find the Equations, Then Solve

• Line 1: slope , so
• Line 2: slope , so
• Set equal: , ,

Slopes Decide Whether Lines Meet

• Different slopes → lines cross once
• Same slope, different intercept → parallel, never
• Same slope and intercept → same line

A Parallel Pair of Lines

Line through and ; line through and .

• Both have slope : and
• Setting equal gives — no intersection

Sort These: Intersect or Parallel?

For each pair of lines, compare slopes and sort:

1. Line through vs line through
2. Line through vs line through

Quick Check: Where Do They Meet?

A line through and , and the line .

Find where they intersect.

The Algebra Says 8.5 Notebooks

A student buys notebooks ($3) and pens ($1), 3 items for $20.

• and give

Can you buy 8.5 notebooks?

Interpret, Then Validate Two Ways

After solving, always:

• Interpret: state the answer in a full sentence
• Check the algebra: both equations hold
• Check the context: is the answer possible?

Chickens and Cows on a Farm

20 heads and 56 legs. How many of each?

• and
• Solve: 12 chickens, 8 cows
• Both counts whole and positive — makes sense

Two Kinds of No Solution

• Contextual: the math answer can't happen (8.5 notebooks)
• Algebraic: the equations contradict, like

Both mean the constraints can't be met together.

A Savings Comparison Over Weeks

Aisha has $40, saves $15/week. Ben has $100, saves $5/week.

• and
• Equal at : both have $130

Solve, Interpret, and Then Validate

A store sells juice ($2) and water ($1). A customer buys 7 bottles for $11.

Solve, write the answer in a sentence, and check it makes sense.

Commit: Write the Two Equations

A theater charges $12 adults, $7 students. 10 tickets cost $95.

On your own, write the two equations. Don't solve.

Your Turn: The Full Cycle

A store sells T-shirts ($15) and hats ($8). A customer buys 5 items for $54.

Set up, solve, interpret in a sentence, and validate.

Three Common Mistakes to Avoid

Two unknowns need two equations — one isn't enough
Answer in words: "7 adults," not just "x = 7"
Check the answer could really happen

Coming Up Next: High School Modeling

In high school, this same six-step cycle grows: more variables, more constraints, and questions about which solutions are truly viable in a real situation.