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Solve Real-World
System Problems

Learn to translate everyday scenarios into two linear equations and find the point where they meet: the one solution that satisfies both constraints at once.

8th Grade Math

What Is a System of Equations?

A system of linear equations is a set of two (or more) equations that share the same variables. You are looking for values of those variables that make every equation true at the same time.

Equation 1
Describes one constraint or relationship between the unknowns.
Equation 2
Describes a second, different constraint on the same unknowns.
Solution
The ordered pair \((x, y)\) that satisfies both equations simultaneously.

Visualizing Two Lines

x y 1 2 3 4 1 2 3
\(y = x + 1\)
\(y = -\frac{1}{2}x + 4\)
\((2, 3)\)

Why Real-World Problems?

Systems of equations appear every time two quantities are connected by two different rules. Learning to translate a word problem into algebra is the key skill.

Money
Mixing ticket prices, comparing phone plans, splitting costs.
Quantities
Combining items that total a known amount or weight.
Rates
Speed, distance, time: two travelers meeting or catching up.

The Standard Form of a System

Every real-world system problem boils down to writing two equations like these from the information given, then solving for \(x\) and \(y\).

Three Solution Methods

Graphing Plot both lines, find intersection Substitution Solve one eq. for a variable, plug in Elimination Add/subtract eqs. to cancel a variable

Substitution vs. Elimination

Substitution Elimination
Best when One variable is already isolated Coefficients are easy to match
First step Solve one eq. for \(x\) or \(y\) Multiply to match coefficients
Then Plug expression into other eq. Add or subtract equations
Risk Messy fractions Sign errors when subtracting

Step-by-Step: Solving a Word Problem

1
Define variables. Choose letters to represent the unknowns you are asked to find.
2
Write two equations. Translate each sentence that links the unknowns into an equation.
3
Solve the system. Use substitution or elimination to find \(x\) and \(y\).
4
Answer in context. State the answer using the original units and check it in both equations.

Example 1: Ticket Sales

A school sold 200 tickets for a play. Adult tickets cost \(\$8\) and student tickets cost \(\$5\). Total revenue was \(\$1,180\). How many of each type were sold?

Solving the Ticket Problem

1
From Eq. 1: \(s = 200 - a\)
2
Substitute into Eq. 2: \(8a + 5(200 - a) = 1180\)
3
Simplify: \(8a + 1000 - 5a = 1180 \Rightarrow 3a = 180 \Rightarrow a = 60\)
4
Back-substitute: \(s = 200 - 60 = 140\)

Example 2: Phone Plans

Plan A charges a \(\$30\) monthly fee plus \(\$0.10\) per text. Plan B charges \(\$20\) per month plus \(\$0.15\) per text. After how many texts do the plans cost the same?

texts cost ($) 100 200 300 30 40 50 60 Plan A Plan B
\((200, \$50)\)

Solving the Phone Plan Problem

1
Set the costs equal: \(30 + 0.10t = 20 + 0.15t\)
2
Subtract \(0.10t\) from both sides: \(30 = 20 + 0.05t\)
3
Subtract 20: \(10 = 0.05t\)
4
Divide: \(t = \dfrac{10}{0.05} = 200\) texts

Example 3: Elimination Method

A store sells notebooks for \(\$3\) each and pens for \(\$2\) each. Mia bought 7 items and spent \(\$17\). How many of each did she buy?

Common Mistakes to Avoid

✕ One Equation Only
Writing just one equation and guessing the second. Every system needs two distinct equations.
✕ Swapped Variables
Assigning \(x\) to one quantity in Eq. 1 but accidentally switching it in Eq. 2.
✕ Forgetting Units
Mixing dollars and cents, or minutes and hours, within the same equation.

Method Summary

Feature Graphing Substitution Elimination
Accuracy Approximate (visual) Exact Exact
Speed Slow Medium Fast (integer coefficients)
Best for Visualizing \(y = \ldots\) form Standard form
Real-world use Estimating break-even Price/quantity problems Mixture/combination problems

Question 1 of 5

A bakery sells muffins for \(\$3\) and cookies for \(\$2\). A customer buys 10 items and spends \(\$24\). How many muffins did the customer buy?

Question 2 of 5

Two numbers have a sum of 50 and a difference of 14. What is the larger number?

Question 3 of 5

Gym A charges a \(\$50\) sign-up fee plus \(\$20/\text{month}\). Gym B charges no sign-up fee but \(\$30/\text{month}\). After how many months do they cost the same?

Question 4 of 5

Which system of equations models this problem? "A garden has roses and tulips. There are 30 flowers total. There are twice as many tulips as roses."

Question 5 of 5

A boat travels 60 km downstream in 3 hours and 60 km upstream in 5 hours. What is the speed of the current?