Two-Variable Equations - Sixth Grade
Complete Notes & Formulas
1. What are Two-Variable Equations?
Definition
A two-variable equation has TWO variables
usually x and y
Solutions are written as ORDERED PAIRS (x, y)
Standard Form
ax + by = c
where a, b, and c are constants
Examples
• y = 2x + 3 (two variables: x and y)
• 3x + 2y = 12 (two variables: x and y)
• y = 5x (two variables: x and y)
2. Ordered Pairs (x, y)
What is an Ordered Pair?
(x, y)
First number (x): x-coordinate (horizontal position)
Second number (y): y-coordinate (vertical position)
ORDER MATTERS! (2, 3) ≠ (3, 2)
Does (x, y) Satisfy an Equation?
Step 1: Substitute x-value into equation
Step 2: Substitute y-value into equation
Step 3: Check if both sides are equal
Step 4: If equal → YES, if not equal → NO
Example: Does (3, 9) satisfy y = 2x + 3?
Given: (x, y) = (3, 9)
x = 3, y = 9
Substitute:
9 = 2(3) + 3
9 = 6 + 3
9 = 9 ✓
Answer: YES, (3, 9) satisfies the equation!
3. Independent and Dependent Variables
Definitions
| Variable Type | Meaning | Usually | Also Called |
|---|---|---|---|
| Independent | The INPUT - what you CHANGE | x | Input, cause |
| Dependent | The OUTPUT - what DEPENDS on input | y | Output, effect |
Independent → Dependent
Input → Output
Cause → Effect
x → y
Example: Cost of Apples
Situation: Apples cost $2 each. Total cost = 2 × number of apples
Independent variable (x): Number of apples (YOU choose)
Dependent variable (y): Total cost (DEPENDS on number)
Equation: y = 2x
4. Creating Tables for Two-Variable Equations
Steps to Complete a Table
Step 1: Choose values for x (independent variable)
Step 2: Substitute each x-value into the equation
Step 3: Calculate the corresponding y-value
Step 4: Write as ordered pair (x, y)
Example: Complete table for y = 3x + 1
| x | y = 3x + 1 | y | Ordered Pair |
|---|---|---|---|
| 0 | 3(0) + 1 | 1 | (0, 1) |
| 1 | 3(1) + 1 | 4 | (1, 4) |
| 2 | 3(2) + 1 | 7 | (2, 7) |
| 3 | 3(3) + 1 | 10 | (3, 10) |
5. Writing Equations from Tables
Steps
Step 1: Look for a pattern in the y-values
Step 2: Find the relationship between x and y
Step 3: Check: Does y equal something times x?
Step 4: Check: Is there a constant added or subtracted?
Step 5: Write the equation
Example: Find equation from table
| x | y |
|---|---|
| 1 | 5 |
| 2 | 10 |
| 3 | 15 |
| 4 | 20 |
Pattern:
• When x = 1, y = 5 (5 × 1)
• When x = 2, y = 10 (5 × 2)
• When x = 3, y = 15 (5 × 3)
• y is always 5 times x
Equation: y = 5x
6. Graphing Two-Variable Equations
The Coordinate Plane
x-axis: Horizontal line (left-right)
y-axis: Vertical line (up-down)
Origin: Where axes meet (0, 0)
Ordered pair (x, y): Point on the plane
Steps to Graph
Step 1: Create a table of values (at least 3 points)
Step 2: Plot each ordered pair (x, y)
Step 3: Connect the points with a straight line
Step 4: Extend the line with arrows on both ends
Visual: Plotting Points
Example: Graph y = x + 2
Points: (0, 2), (1, 3), (2, 4) all lie on the line y = x + 2
7. Finding Values Using Two-Variable Equations
Two Types of Problems
Type 1: Given x, find y (most common)
Type 2: Given y, find x (solve for x)
Example 1: Given x = 4, find y in y = 3x - 2
y = 3x - 2
y = 3(4) - 2
y = 12 - 2
y = 10
Answer: When x = 4, y = 10
Example 2: Given y = 15, find x in y = 5x
y = 5x
15 = 5x
15 ÷ 5 = x
x = 3
Answer: When y = 15, x = 3
8. Two-Variable Equation Word Problems
Steps
Step 1: Identify the two variables
Step 2: Determine which is independent and dependent
Step 3: Write the equation
Step 4: Solve for the unknown
Step 5: Check your answer
Example: Movie Tickets
Problem: Movie tickets cost $8 each. Write an equation for total cost based on number of tickets. Then find the cost for 5 tickets.
Step 1: Identify variables
x = number of tickets (independent)
y = total cost (dependent)
Step 2: Write equation
y = 8x
Step 3: Find cost for 5 tickets (x = 5)
y = 8(5)
y = 40
Answer: y = 8x; Cost for 5 tickets is $40
9. Interpreting Graphs
What to Look For
• Title: What does the graph show?
• x-axis label: What is the independent variable?
• y-axis label: What is the dependent variable?
• Points on line: What are the ordered pairs?
• Slope: Is it going up or down?
Reading Values from Graph
To find y given x:
1. Find x on horizontal axis
2. Go straight up to the line
3. Read y-value from vertical axis
To find x given y:
1. Find y on vertical axis
2. Go straight across to the line
3. Read x-value from horizontal axis
Quick Reference: Two-Variable Equations
| Concept | Key Points | Example |
|---|---|---|
| Ordered Pair | (x, y) - order matters! | (3, 5) |
| Independent Variable | Input (x) - what you change | Number of items |
| Dependent Variable | Output (y) - depends on x | Total cost |
| Graph | Points connected by line | Straight line |
💡 Important Tips to Remember
✓ Ordered pairs: (x, y) - order matters!
✓ Independent variable (x) is the INPUT (what you choose)
✓ Dependent variable (y) is the OUTPUT (what depends)
✓ To check solution: substitute both x and y into equation
✓ Tables help organize values
✓ Graphs show all solutions visually
✓ Two-variable equations have infinitely many solutions
✓ Linear equations graph as straight lines
✓ x-axis is horizontal, y-axis is vertical
✓ Always label axes on graphs!
🧠 Memory Tricks & Strategies
Ordered Pairs:
"X comes before Y in alphabet - so x comes first in the pair!"
Independent vs Dependent:
"Independent is IN-put, Dependent is D-output (depends on input)!"
Axes:
"X-axis goes across (X looks like a cross), Y-axis goes up (Y reaches for the skY)!"
Graphing:
"Make a table, plot the dots, connect with line - then you've got it!"
Reading Graphs:
"Start at x, go up or across - find the point where the line does cross!"
Master Two-Variable Equations! (x, y) 📊 📈
Remember: Independent INPUT → Dependent OUTPUT!