How to Explain Systems of Equations to Eighth Graders

"I bought 3 apples and 2 oranges for $5. My friend bought 2 apples and 4 oranges for $6. How much does each fruit cost?"

One equation isn't enough here—you need TWO. Welcome to systems of equations, where two pieces of information work together to find two unknowns.

Why Systems of Equations Matter

Systems of equations appear everywhere:

• Business problems (cost and revenue)
• Science experiments (two variables)
• Real-world planning and budgeting
• Foundation for linear algebra
• Essential for physics and engineering

What IS a System of Equations?

Definition

A system of equations is two or more equations with the same variables.

Equation 1:  2x + y = 10
Equation 2:  x - y = 2

The Goal

Find values that make BOTH equations true at the same time.

Solution: x = 4, y = 2

Check Equation 1: 2(4) + 2 = 8 + 2 = 10 ✓
Check Equation 2: 4 - 2 = 2 ✓

Visualizing the Solution

Each equation represents a line. The solution is where the lines intersect.

        y
        |
      6 +        /
        |      /  Equation 1
      4 +    /
        |  /----* (4, 2) Solution!
      2 +/
        |     \
      0 +---+---+---+---+
        0   2   4   6   x
              \
               Equation 2

Three Solution Types

One Solution

Lines intersect at exactly one point.

        y
        |   \  /
        |    \/
        |    /\
        +---+---+---
            x

Most systems have one solution.

No Solution

Lines are parallel—they never meet.

        y
        |   /  /
        |  /  /
        | /  /
        +---+---+---
            x

When solving, you get a false statement like 0 = 5.

Infinite Solutions

Lines are the same—every point on the line is a solution.

        y
        |   /
        |  / (same line)
        | /
        +---+---+---
            x

When solving, you get a true statement like 0 = 0.

Method 1: Graphing

Steps

1. Write each equation in slope-intercept form (y = mx + b)
2. Graph both lines
3. Find the intersection point
4. Check the solution in both equations

Example

Solve by graphing:

y = 2x - 1
y = -x + 5

Graph both lines:

Line 1: y = 2x - 1
  - y-intercept: (0, -1)
  - slope: 2 (up 2, right 1)

Line 2: y = -x + 5
  - y-intercept: (0, 5)
  - slope: -1 (down 1, right 1)

        y
      6 +
        |\
      4 + \
        |  \    /
      2 +   *--/--- (2, 3)
        |     /
      0 +---+/--+---
       -2   0  2  4   x
        |  /
     -2 + /

Solution: (2, 3)

Check:

• y = 2(2) - 1 = 3 ✓
• y = -(2) + 5 = 3 ✓

Limitations of Graphing

• Hard to read exact answers when not integers
• Time-consuming
• Best for visual understanding, not precision

Method 2: Substitution

When to Use

Use substitution when one equation is already solved for a variable (or easily can be).

Steps

1. Solve one equation for one variable
2. Substitute that expression into the other equation
3. Solve for the remaining variable
4. Substitute back to find the other variable
5. Check in BOTH original equations

Example 1: One Variable Already Isolated

y = 3x - 5
2x + y = 10

Step 1: y is already isolated in Equation 1.

Step 2: Substitute (3x - 5) for y in Equation 2:

2x + (3x - 5) = 10

Step 3: Solve:

2x + 3x - 5 = 10
5x - 5 = 10
5x = 15
x = 3

Step 4: Substitute back:

y = 3(3) - 5 = 9 - 5 = 4

Solution: (3, 4)

Step 5: Check both:

• y = 3(3) - 5 = 4 ✓
• 2(3) + 4 = 10 ✓

Example 2: Need to Isolate First

x + 2y = 7
3x - y = 11

Step 1: Isolate x in Equation 1:

x = 7 - 2y

Step 2: Substitute into Equation 2:

3(7 - 2y) - y = 11

Step 3: Solve:

21 - 6y - y = 11
21 - 7y = 11
-7y = -10
y = 10/7

Step 4: Substitute back:

x = 7 - 2(10/7) = 7 - 20/7 = 49/7 - 20/7 = 29/7

Solution: (29/7, 10/7)

Method 3: Elimination

When to Use

Use elimination when coefficients can easily be made opposites.

Steps

1. Arrange equations with like terms aligned
2. Multiply one or both equations so one variable has opposite coefficients
3. Add the equations to eliminate that variable
4. Solve for the remaining variable
5. Substitute back to find the other variable
6. Check in BOTH original equations

Example 1: Opposite Coefficients Already There

2x + y = 7
3x - y = 8

Step 2: Already have +y and -y!

Step 3: Add the equations:

  2x + y = 7
+ 3x - y = 8
-----------
  5x     = 15

Step 4: Solve:

x = 3

Step 5: Substitute back (into either equation):

2(3) + y = 7
6 + y = 7
y = 1

Solution: (3, 1)

Example 2: Need to Create Opposites

3x + 2y = 16
5x + 3y = 25

Goal: Make y-coefficients opposites.

Step 2: Multiply:

• Equation 1 by 3: 9x + 6y = 48
• Equation 2 by -2: -10x - 6y = -50

Step 3: Add:

   9x + 6y = 48
+ -10x - 6y = -50
--------------
   -x       = -2

Step 4: x = 2

Step 5: Substitute:

3(2) + 2y = 16
6 + 2y = 16
2y = 10
y = 5

Solution: (2, 5)

Example 3: Special Case - No Solution

2x + 4y = 10
x + 2y = 8

Multiply Equation 2 by -2:

-2x - 4y = -16

Add to Equation 1:

  2x + 4y = 10
+ -2x - 4y = -16
--------------
  0 = -6  ← FALSE!

No solution. The lines are parallel.

Example 4: Special Case - Infinite Solutions

x + y = 5
2x + 2y = 10

Multiply Equation 1 by -2:

-2x - 2y = -10

Add:

  2x + 2y = 10
+ -2x - 2y = -10
--------------
  0 = 0  ← TRUE!

Infinite solutions. Same line!

Choosing a Method

Situation	Best Method
One variable isolated	Substitution
Coefficients are opposites	Elimination
Coefficients easy to make opposites	Elimination
Need visual understanding	Graphing
Equations in y = mx + b form	Graphing

Word Problems with Systems

The Translation Process

1. Identify the two unknowns (assign variables)
2. Write two different equations using the information
3. Solve the system
4. Answer the question with units

Example: Fruit Problem

"I bought 3 apples and 2 oranges for $5.00. My friend bought 2 apples and 4 oranges for $6.00. Find the price of each."

Variables:

• a = price of one apple
• o = price of one orange

Equations:

3a + 2o = 5   (My purchase)
2a + 4o = 6   (Friend's purchase)

Solve using elimination:

Multiply Equation 1 by -2:

-6a - 4o = -10
2a + 4o = 6
-----------
-4a = -4
a = 1

Substitute:

3(1) + 2o = 5
2o = 2
o = 1

Answer: Apples cost $1.00 each, oranges cost $1.00 each.

Example: Distance Problem

"Two cars leave from the same point. One travels north at 60 mph, the other south at 40 mph. After how many hours will they be 300 miles apart?"

This actually only needs one equation! But here's a systems approach:

Variables:

• t = time traveled
• d₁ = distance of car 1
• d₂ = distance of car 2

Equations:

d₁ = 60t
d₂ = 40t
d₁ + d₂ = 300

Substitute:

60t + 40t = 300
100t = 300
t = 3 hours

Example: Mixture Problem

"A chemist needs 100 mL of a 40% acid solution. She has 30% and 60% solutions. How much of each should she mix?"

Variables:

• x = mL of 30% solution
• y = mL of 60% solution

Equations:

x + y = 100           (total volume)
0.30x + 0.60y = 40    (total acid)

Solve:
From Equation 1: x = 100 - y

Substitute:

0.30(100 - y) + 0.60y = 40
30 - 0.30y + 0.60y = 40
30 + 0.30y = 40
0.30y = 10
y = 33.33 mL

x = 100 - 33.33 = 66.67 mL

Answer: About 67 mL of 30% solution and 33 mL of 60% solution.

Hands-On Activities

Graphing Calculator Exploration

Enter equations in Y= and find the intersection using the calculator's intersect feature.

System Scavenger Hunt

Give students different systems. Each solution is a clue leading to the next problem.

Real-Life Systems

Research actual scenarios:

• Cell phone plans (monthly fee + per-minute cost)
• At what point is Plan A cheaper than Plan B?

Error Analysis Challenge

Present "solved" systems with errors. Students find and fix mistakes.

Create Your Own Story Problem

Give students a solution (x = 3, y = 5) and have them write a word problem that produces this system.

Common Mistakes and How to Fix Them

Mistake 1: Forgetting to Substitute Back

Wrong: Found x = 3, stopped there.

Fix: You need BOTH variables. Always find x AND y.

Mistake 2: Sign Errors in Elimination

Wrong:

  3x + 2y = 7
- 3x - y = 4
-----------
  0 + y = 3   ← Should be 3y!

Fix: Be extra careful with signs. Subtracting a negative becomes adding.

Mistake 3: Not Checking Both Equations

Wrong: Only checked in one equation.

Fix: The solution must work in BOTH. Always check both.

Mistake 4: Distributing Errors in Substitution

Wrong:

If y = 3x - 2, then:
2x + y = 2x + 3x - 2 = 5x - 2 ✓
But 3(y) = 3(3x - 2) = 9x - 2 ✗

Fix: Distribute to ALL terms: 3(3x - 2) = 9x - 6

Mistake 5: Misinterpreting Special Cases

Fix:

• 0 = 0 means INFINITELY MANY solutions
• 0 = 5 means NO solution
• Neither means ONE solution

Practice Ideas for Home

Daily System Practice

Solve one system each day using different methods:

• Day 1: Graph it
• Day 2: Substitution
• Day 3: Elimination

Real-World Systems

Find systems in daily life:

• "Two shirts and three pants cost $85. One shirt and two pants cost $50."
• "Adult tickets are $12, child tickets are $8. 50 people paid $520 total."

Speed Challenge

Time yourself solving systems. Track improvement over time.

Connecting to Future Concepts

Three Variables (Algebra 2)

x + y + z = 6
2x - y + z = 3
x + 2y - z = 3

Same methods, just more steps!

Matrices

Systems can be represented as matrices:

[2  1] [x]   [7]
[3 -1] [y] = [8]

Linear Programming

Businesses use systems of inequalities to maximize profit:

2x + 3y ≤ 12
x + y ≤ 5
Maximize: P = 5x + 4y

Physics Applications

Force equilibrium, circuit analysis, and many physics problems require systems.

The Bottom Line

Systems of equations solve problems that single equations can't—situations with two unknowns requiring two pieces of information.

The three methods each have their strengths:

• Graphing builds visual intuition
• Substitution is straightforward when a variable is isolated
• Elimination is powerful when coefficients align

Students should be comfortable with all three and know when each works best. The skill of finding where two constraints intersect is fundamental to science, business, engineering, and everyday problem-solving.

When students can confidently solve systems, they've mastered a crucial algebraic skill that will serve them throughout their mathematical journey.