Complete lesson: slope-intercept form, graphing, interpretation, and 10 practice questions with solutions.
📖 What Is a Linear Equation in Two Variables?
A linear equation in two variables is an equation that can be written in the form:
y = mx + b
where m is the slope and b is the y-intercept. This is called slope-intercept form.
📌 Key Concept
The graph of a linear equation is a straight line. The slope (m) tells you how steep the line is, and the y-intercept (b) tells you where it crosses the y-axis.
📐 Slope (m)
The slope measures the steepness of a line. It is calculated as:
m = (y₂ − y₁) / (x₂ − x₁)
Positive slope: Line goes up from left to right
Negative slope: Line goes down from left to right
Zero slope: Horizontal line
Undefined slope: Vertical line
📐 y-Intercept (b)
The y-intercept is the point where the line crosses the y-axis (where x = 0).
⚡ 3-Step Strategy
STEP 1
Identify the Form
Is it in slope-intercept form (y = mx + b)? If not, rearrange it.
STEP 2
Extract Information
Identify the slope (m) and y-intercept (b) from the equation.
STEP 3
Graph or Interpret
Plot the y-intercept, use slope to find another point, draw the line.
💡 Key Strategy
Always convert to y = mx + b form first. Then the slope and y-intercept are immediately visible.
📝 Solved Examples
Study these examples carefully. Each shows the step-by-step solution process.
EXAMPLE 1
Find the slope and y-intercept: y = 3x + 5
Step 1: Compare to y = mx + b
Step 2: m = 3 (slope), b = 5 (y-intercept)
✅ Slope = 3, y-intercept = 5
💡 Tip: The line rises 3 units for every 1 unit it moves right.
EXAMPLE 2
Find the slope and y-intercept: 2x + y = 6
Step 1: Solve for y: y = −2x + 6
Step 2: m = −2, b = 6
✅ Slope = −2, y-intercept = 6
💡 Tip: A negative slope means the line goes down from left to right.
EXAMPLE 3
Find the slope between points (2, 3) and (5, 9)
Step 1: Use the slope formula: m = (y₂ − y₁) / (x₂ − x₁)
Step 2: m = (9 − 3) / (5 − 2) = 6 / 3 = 2
✅ Slope = 2
💡 Tip: Positive slope means the line is increasing.
EXAMPLE 4
Write the equation of a line with slope = 4 and y-intercept = −2
Step 1: Use y = mx + b
Step 2: Substitute m = 4, b = −2 → y = 4x − 2
✅ y = 4x − 2
💡 Tip: Always write the equation in the form y = mx + b.
EXAMPLE 5
Find the x-intercept of the line: y = 2x − 6
Step 1: To find x-intercept, set y = 0
Step 2: 0 = 2x − 6 → 2x = 6 → x = 3
✅ x-intercept = 3
💡 Tip: The x-intercept is where the line crosses the x-axis (y = 0).
🧪 Practice Questions
Solve each problem. Click "Show Answer" to see the full solution.
Question 1
Find the slope and y-intercept: y = 4x − 7
A) Slope = −4, y-intercept = 7
B) Slope = 4, y-intercept = −7
C) Slope = 7, y-intercept = 4
D) Slope = −7, y-intercept = 4
✓ Answer: B
y = 4x − 7 → m = 4, b = −7
📝 Solution: Compare to y = mx + b → m = 4, b = −7
Question 2
Find the slope between points (1, 4) and (3, 10)
A) 2
B) 3
C) 4
D) 5
✓ Answer: C
m = (10 − 4) / (3 − 1) = 6 / 2 = 3
📝 Solution: m = (10 − 4)/(3 − 1) = 6/2 = 3
Question 3
What is the y-intercept of 3x + y = 9?
A) 3
B) 6
C) 8
D) 9
✓ Answer: D
3x + y = 9 → y = −3x + 9 → b = 9
📝 Solution: y = −3x + 9 → y-intercept = 9
Question 4
Write the equation of a line with slope = −2 and y-intercept = 3
A) y = 2x + 3
B) y = −2x + 3
C) y = 3x − 2
D) y = −2x − 3
✓ Answer: B
y = mx + b → y = −2x + 3
📝 Solution: Substitute m = −2, b = 3 → y = −2x + 3
Question 5
What is the slope of the line passing through (0, 2) and (4, 14)?
A) 2
B) 3
C) 4
D) 5
✓ Answer: C
m = (14 − 2) / (4 − 0) = 12 / 4 = 3
📝 Solution: m = (14 − 2)/(4 − 0) = 12/4 = 3
Question 6
Find the x-intercept: y = 5x − 20
A) x = 2
B) x = 4
C) x = 5
D) x = 10
✓ Answer: B
Set y = 0: 0 = 5x − 20 → 5x = 20 → x = 4
📝 Solution: 0 = 5x − 20 → 5x = 20 → x = 4
Question 7
Which equation has slope 5 and y-intercept −2?
A) y = 5x + 2
B) y = −5x + 2
C) y = 5x − 2
D) y = −5x − 2
✓ Answer: C
y = mx + b → m = 5, b = −2 → y = 5x − 2
📝 Solution: y = 5x − 2
Question 8
Find the slope between (−2, 1) and (3, 6)
A) 1
B) 2
C) 3
D) 4
✓ Answer: A
m = (6 − 1) / (3 − (−2)) = 5 / 5 = 1
📝 Solution: m = (6 − 1)/(3 + 2) = 5/5 = 1
Question 9
What is the slope of y = 7?
A) 0
B) 1
C) 7
D) Undefined
✓ Answer: A
y = 7 is a horizontal line → slope = 0
📝 Solution: y = 7 → y = 0x + 7 → slope = 0
Question 10
Which line is parallel to y = 2x + 3?
A) y = −2x + 5
B) y = 2x − 1
C) y = 3x + 2
D) y = −3x − 4
✓ Answer: B
Parallel lines have the same slope. y = 2x + 3 has slope 2. y = 2x − 1 also has slope 2.
📝 Solution: Slope = 2 → y = 2x − 1 is parallel.
🎉 WELL DONE!
You've completed the Linear Equations in Two Variables lesson. You now know how to work with slopes, intercepts, and graph interpretation.