Writing Linear Equations
\( y = mx + b \)
Learn to write the equation of a line from a graph, table, or real-world context using slope-intercept form.
Pre-Algebra
What Is a Linear Equation?
A linear equation describes a relationship between two variables that forms a straight line when graphed. Every point on the line is a solution to the equation.
The Two Key Ingredients
\( m \) : Slope
The rate of change: how much \( y \) increases for every 1-unit increase in \( x \). Computed as \( m = \frac{\text{rise}}{\text{run}} \).
\( b \) : Y-Intercept
The point where the line crosses the y-axis: the value of \( y \) when \( x = 0 \).
Why Does This Matter?
Linear equations model countless real-world situations: earning money over time, tracking distance traveled, measuring growth, or comparing phone plans. Writing the equation lets you predict values you haven't observed yet.
★ Predict
Find future or missing values by plugging in \( x \).
★ Compare
See which option grows faster by comparing slopes.
★ Communicate
Express a pattern in one concise equation.
The Slope-Intercept Form
Reading \( m \) and \( b \) from a Graph
Positive vs Negative Slope
Positive Slope
Line goes up from left to right. \( m > 0 \). Example: \( y = 2x + 3 \).
Negative Slope
Line goes down from left to right. \( m < 0 \). Example: \( y = -x + 5 \).
Writing from a Table
| \( x \) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \( y \) | 5 | 8 | 11 | 14 | 17 |
1
Find the y-intercept: when \( x = 0 \), \( y = \) 5, so \( b = 5 \).
2
Find the slope: each time \( x \) goes up by 1, \( y \) goes up by 3. So \( m = \frac{3}{1} = 3 \).
3
Write the equation: \( y = 3x + 5 \).
Writing from a Real-World Context
1
Identify the rate (slope): $12 per pizza means \( m = 12 \).
2
Identify the starting value (y-intercept): $8 delivery fee is paid even with 0 pizzas, so \( b = 8 \).
3
Write the equation: \( y = 12x + 8 \).
How to Apply the Method
1
Identify the source: Are you given a graph, a table, or a word problem?
2
Find \( b \) (y-intercept): Look for the value of \( y \) when \( x = 0 \). On a graph, it's where the line crosses the y-axis.
3
Find \( m \) (slope): Pick two points and calculate \( m = \frac{y_2 - y_1}{x_2 - x_1} \). In a context, identify the "per" rate.
4
Substitute into \( y = mx + b \): Plug in your values of \( m \) and \( b \).
5
Verify: Test a known point from the original data to make sure it satisfies your equation.
Example 1: From a Graph
Example 2: From a Table
| \( x \) | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| \( y \) | 1 | 5 | 9 | 13 |
1
Find the slope: \( m = \frac{5 - 1}{4 - 2} = \frac{4}{2} = 2 \).
2
Find \( b \): \( x = 0 \) isn't in the table, so use a known point. Plug \((2, 1)\) into \( y = 2x + b \): \( 1 = 2(2) + b \), so \( b = -3 \).
3
Write the equation: \( y = 2x - 3 \).
4
Verify: Check \((6, 9)\): \( 2(6) - 3 = 9 \). ✓
Example 3: Word Problem
1
Starting amount (y-intercept): 200 gallons at time 0, so \( b = 200 \).
2
Rate of change (slope): 25 gallons per minute, so \( m = 25 \).
3
Equation: \( y = 25x + 200 \).
4
Verify: After 4 minutes: \( y = 25(4) + 200 = 300 \) gallons. ✓
Common Mistakes to Avoid
✕ Swapping Rise and Run
Computing \( \frac{x_2 - x_1}{y_2 - y_1} \) instead of \( \frac{y_2 - y_1}{x_2 - x_1} \). Remember: rise (y) over run (x).
✕ Forgetting the Sign
A line going downhill has a negative slope. If you get a positive number for a downhill line, check your subtraction order.
✕ Wrong Y-Intercept
Reading the x-intercept (where the line crosses the x-axis) instead of the y-intercept (where it crosses the y-axis).
Summary: Three Sources
| Source | How to find \( m \) | How to find \( b \) |
|---|---|---|
| Graph | Count \(\frac{\text{rise}}{\text{run}}\) between two points | Read where line crosses y-axis |
| Table | Compute \( \frac{\Delta y}{\Delta x} \) from two rows | Find \( y \) when \( x = 0 \), or solve for \( b \) |
| Context | Identify the "per" rate | Identify the starting/fixed amount |
Question 1 of 5
A line passes through \((0, 3)\) and \((2, 7)\). What is the equation of the line?
The slope is \( m = \frac{7 - 3}{2 - 0} = \frac{4}{2} = 2 \), and the y-intercept is \( b = 3 \) (the y-value when \( x = 0 \)). So the equation is \( y = 2x + 3 \).
Question 2 of 5
A table shows: when \( x = 1 \), \( y = 10 \); when \( x = 3 \), \( y = 4 \). What is the slope?
The slope is \( m = \frac{4 - 10}{3 - 1} = \frac{-6}{2} = -3 \). The negative sign means the line goes downhill from left to right.
Question 3 of 5
A gym charges a $30 sign-up fee plus $15 per month. Which equation represents the total cost \( y \) after \( x \) months?
The "per month" rate is the slope: \( m = 15 \). The sign-up fee is a one-time starting cost: \( b = 30 \). So \( y = 15x + 30 \).
Question 4 of 5
The equation of a line is \( y = -4x + 12 \). What is the y-intercept?
In \( y = mx + b \), the y-intercept is the value of \( b \). Here, \( b = 12 \). The slope is \( m = -4 \), which is a separate value.
Question 5 of 5
A line has slope \( \frac{1}{2} \) and passes through the point \((4, 5)\). What is the equation?
Start with \( y = \frac{1}{2}x + b \). Plug in the point \((4, 5)\): \( 5 = \frac{1}{2}(4) + b \), so \( 5 = 2 + b \), which gives \( b = 3 \). The equation is \( y = \frac{1}{2}x + 3 \).