Two-Variable Equations - Seventh Grade
Variables, Tables, Graphing & Linear Equations
1. Independent and Dependent Variables
Definitions
Independent Variable (INPUT)
The variable that YOU control or choose
→ It does NOT depend on anything else
→ Usually represented by x
→ Plotted on the HORIZONTAL axis (x-axis)
Dependent Variable (OUTPUT)
The variable that DEPENDS on the independent variable
→ Its value changes based on the independent variable
→ Usually represented by y
→ Plotted on the VERTICAL axis (y-axis)
Example
Situation: The cost of apples depends on how many pounds you buy. Apples cost $3 per pound.
Independent Variable: Number of pounds (x)
→ You choose how many pounds to buy
Dependent Variable: Total cost (y)
→ The cost depends on pounds bought
Equation: y = 3x
x is independent, y is dependent
How to Identify
Ask yourself: "What depends on what?"
• What causes the change? → Independent
• What is affected by the change? → Dependent
2. What is a Two-Variable Equation?
Definition
A two-variable equation has TWO variables
usually x and y
Examples: y = 2x + 3, y = 5x, 2x + y = 10
Standard Forms
y = mx + b
(Slope-intercept form)
Ax + By = C
(Standard form)
Key Point
Two-variable equations have INFINITELY MANY solutions
Each solution is an ordered pair (x, y)
3. Finding Values Using Two-Variable Equations
Strategy
Step 1: Identify which variable you know
Step 2: Substitute that value into the equation
Step 3: Solve for the unknown variable
Example 1: Finding y when x is given
Equation: y = 3x + 2
Find y when x = 4
Step 1: Substitute x = 4
y = 3(4) + 2
Step 2: Simplify
y = 12 + 2
y = 14
Solution: When x = 4, y = 14
Ordered pair: (4, 14)
Example 2: Finding x when y is given
Equation: y = 2x − 5
Find x when y = 7
Substitute y = 7:
7 = 2x − 5
12 = 2x
x = 6
Solution: When y = 7, x = 6
4. Completing Tables for Two-Variable Relationships
Strategy
Use the equation to find missing values
Substitute known values and solve for unknowns
Example
Complete the table for y = 4x + 1
| x | y |
|---|---|
| 0 | ? |
| 1 | ? |
| 2 | ? |
| 3 | ? |
Solution:
When x = 0: y = 4(0) + 1 = 1
When x = 1: y = 4(1) + 1 = 5
When x = 2: y = 4(2) + 1 = 9
When x = 3: y = 4(3) + 1 = 13
| x | y |
|---|---|
| 0 | 1 |
| 1 | 5 |
| 2 | 9 |
| 3 | 13 |
5. Writing Two-Variable Equations
From Word Problems
Step 1: Identify the independent variable (x)
Step 2: Identify the dependent variable (y)
Step 3: Find the relationship between them
Step 4: Write the equation in the form y = ___
Example 1
Problem: A plumber charges $50 for a house call plus $40 per hour. Write an equation for the total cost.
Step 1: x = number of hours (independent)
Step 2: y = total cost (dependent)
Step 3: Cost = $50 + $40 per hour
Step 4: y = 40x + 50
Equation: y = 40x + 50
Example 2
Problem: Movie tickets cost $12 each. Write an equation for the total cost.
x = number of tickets
y = total cost
Cost = $12 × number of tickets
Equation: y = 12x
6. Graphing Two-Variable Equations
Methods to Graph
Method 1: Make a Table
• Choose several x-values
• Calculate corresponding y-values
• Plot the points
• Connect with a straight line
Method 2: Use Slope and y-intercept
• Find y-intercept (where line crosses y-axis)
• Use slope to find other points
• Draw the line
Example: Graph y = 2x + 1
Method 1: Using a table
| x | y = 2x + 1 | Point (x, y) |
|---|---|---|
| -1 | -1 | (-1, -1) |
| 0 | 1 | (0, 1) |
| 1 | 3 | (1, 3) |
| 2 | 5 | (2, 5) |
Plot these points and connect with a straight line
Method 2: Using slope and y-intercept
y = 2x + 1
Slope (m) = 2 = 2/1 (rise 2, run 1)
y-intercept (b) = 1 (point: 0, 1)
Start at (0, 1), go up 2 and right 1 to get (1, 3)
Draw line through these points
7. Writing Equations from Graphs
Steps for Linear Equations
Step 1: Find the y-intercept (b)
→ Where the line crosses the y-axis
Step 2: Find the slope (m)
→ Pick two points on the line
→ Calculate: m = (y₂ − y₁)/(x₂ − x₁)
Step 3: Write the equation
→ Use form: y = mx + b
Example
A line passes through (0, 3) and (2, 7). Write the equation.
Step 1: y-intercept at (0, 3)
b = 3
Step 2: Find slope using (0, 3) and (2, 7)
m = (7 − 3)/(2 − 0) = 4/2 = 2
Step 3: Write equation
y = mx + b
y = 2x + 3
Equation: y = 2x + 3
8. Interpreting Graphs: Word Problems
What to Look For
Y-intercept: Starting value or initial amount
Slope: Rate of change (how fast y changes)
Positive slope: Increasing relationship
Negative slope: Decreasing relationship
Steeper line: Faster rate of change
Example
Situation: A graph shows Sarah's savings over time. The equation is y = 20x + 50.
Interpretation:
• x represents weeks
• y represents total savings (dollars)
• Slope = 20: Sarah saves $20 per week
• y-intercept = 50: She started with $50
Questions you can answer:
• How much after 5 weeks? y = 20(5) + 50 = $150
• When will she have $250? 250 = 20x + 50 → x = 10 weeks
Quick Reference: Linear Equations
Slope-Intercept Form
y = mx + b
m = slope (rate of change)
b = y-intercept (starting value)
| Task | What to Do |
|---|---|
| Find y given x | Substitute x into equation, solve for y |
| Find x given y | Substitute y into equation, solve for x |
| Graph equation | Make table or use slope and y-intercept |
| Write equation from graph | Find m and b, write y = mx + b |
💡 Important Tips to Remember
✓ Independent variable (x): The input, what you control
✓ Dependent variable (y): The output, depends on x
✓ Two-variable equations: Have infinitely many solutions
✓ Each solution: Is an ordered pair (x, y)
✓ Graph: All solutions form a straight line (for linear equations)
✓ Slope (m): Tells you the rate of change
✓ Y-intercept (b): Where line crosses y-axis (starting value)
✓ To graph: Make a table or use slope and y-intercept
✓ From graph to equation: Find m and b, write y = mx + b
✓ Always check: Does your point satisfy the equation?
🧠 Memory Tricks & Strategies
Independent vs Dependent:
"Independent stands alone, Dependent needs help to be known!"
X and Y Variables:
"X marks the spot you choose, Y is what you win or lose!"
Slope-Intercept Form:
"Y = MX + B, slope and start for you and me!"
Finding Y:
"When X is known, substitute and simplify - that's how you find Y!"
Graphing:
"Start at B, use M to see - where the next point will be!"
Interpreting Slope:
"Slope is the rate, how things relate - positive grows, negative goes!"
Master Two-Variable Equations! 📊 📈
Remember: y = mx + b (slope-intercept form)