Slope-Intercept Form

The slope-intercept form of a linear equation is the most widely used way to write the equation of a line. Once you can recognize this form and pull out the slope and y-intercept, graphing becomes fast and interpreting real-world relationships becomes intuitive.

What Is Slope-Intercept Form?

A linear equation is in slope-intercept form when it is written as:

$y = mx + b$

where:

• $m$ is the slope — the rate of change, or how steep the line is
• $b$ is the y-intercept — the value of $y$ when $x = 0$, i.e., the point where the line crosses the $y$-axis

This form is powerful because both key properties of the line — its steepness and its starting point — are immediately visible.

Identifying Slope and Y-Intercept

To read the slope and y-intercept from an equation already in slope-intercept form, simply match it to the pattern $y = mx + b$.

Example 1: Read $m$ and $b$ directly

Identify the slope and y-intercept of $y = 3x - 7$.

Step 1 — Match to the pattern: Compare $y = 3x - 7$ with $y = mx + b$.

Step 2 — Identify the slope: The coefficient of $x$ is $3$, so $m = 3$.

Step 3 — Identify the y-intercept: The constant term is $-7$, so $b = -7$. The line crosses the $y$-axis at $(0, -7)$.

Example 2: Equation with a fractional slope

Identify the slope and y-intercept of $y = -\frac{2}{5}x + 4$.

Matching to $y = mx + b$:

• Slope: $m = -\frac{2}{5}$ (the line falls 2 units for every 5 units to the right)
• Y-intercept: $b = 4$, so the line crosses the $y$-axis at $(0, 4)$

Writing Equations in Slope-Intercept Form

If you know the slope and y-intercept of a line, you can write its equation by substituting directly into $y = mx + b$.

Example 3: From slope and y-intercept

Write the equation of a line with slope $-2$ and y-intercept $5$.

$y = -2x + 5$

That is it. Plug $m = -2$ and $b = 5$ into the formula.

Example 4: From a graph

Suppose a graph shows a line crossing the $y$-axis at $(0, 3)$ and passing through the point $(4, 7)$.

Step 1 — Read the y-intercept: $b = 3$.

Step 2 — Calculate the slope using the two visible points $(0, 3)$ and $(4, 7)$:

$m = \frac{7 - 3}{4 - 0} = \frac{4}{4} = 1$

Step 3 — Write the equation:

$y = 1x + 3 = x + 3$

Rearranging Equations into Slope-Intercept Form

Not every linear equation starts in the form $y = mx + b$. You often need to solve for $y$ first.

Example 5: Convert from standard form

Rewrite $4x + 2y = 10$ in slope-intercept form.

Step 1 — Isolate the $y$-term: Subtract $4x$ from both sides.

$2y = -4x + 10$

Step 2 — Divide every term by the coefficient of $y$:

$y = -2x + 5$

Now you can read: slope $m = -2$, y-intercept $b = 5$.

Example 6: Convert when $y$-term has a negative coefficient

Rewrite $6x - 3y = 9$ in slope-intercept form.

Step 1 — Subtract $6x$:

$-3y = -6x + 9$

Step 2 — Divide by $-3$:

$y = 2x - 3$

Slope: $m = 2$, y-intercept: $b = -3$.

Graphing from Slope-Intercept Form

The slope-intercept form gives you a ready-made recipe for graphing:

1. Plot the y-intercept $(0, b)$ on the $y$-axis
2. Use the slope to find additional points: from the y-intercept, move up (or down) by the rise and right by the run
3. Draw the line through the points

Example 7: Graph $y = \frac{3}{2}x - 1$

Step 1 — Plot the y-intercept: $(0, -1)$.

Step 2 — Apply the slope: $m = \frac{3}{2}$ means rise 3, run 2. From $(0, -1)$, move up 3 and right 2 to reach $(2, 2)$.

Step 3 — Find a third point for accuracy: From $(2, 2)$, move up 3 and right 2 to reach $(4, 5)$.

Step 4 — Draw the line through $(0, -1)$, $(2, 2)$, and $(4, 5)$.

Example 8: Graph $y = -x + 4$

Step 1 — Identify slope and intercept: $m = -1 = \frac{-1}{1}$, $b = 4$.

Step 2 — Plot $(0, 4)$.

**Step 3 — From $(0, 4)$, move down 1 and right 1 to $(1, 3)$. Repeat to get $(2, 2)$.

Step 4 — Draw the line through the three points. The negative slope means it falls from left to right.

Special Cases

Horizontal lines: An equation like $y = 5$ can be written as $y = 0x + 5$. The slope is $0$ and the y-intercept is $5$. The line is flat.

Lines through the origin: An equation like $y = 3x$ has $b = 0$. The line passes through the origin $(0, 0)$.

No y-intercept form for vertical lines: A vertical line like $x = 2$ cannot be written in slope-intercept form because its slope is undefined.

Real-World Application: HVAC — Heating Cost Estimation

An HVAC company charges a flat service fee plus an hourly rate. A technician’s billing follows the equation:

$y = 65x + 95$

where $x$ is the number of hours worked and $y$ is the total cost in dollars.

• The slope $m = 65$ tells you the hourly rate: $65 per hour
• The y-intercept $b = 95$ tells you the flat service call fee: $95

Using the equation for estimates:

• A 2-hour job costs $y = 65(2) + 95 = 130 + 95 = 225$, so $225
• A 4-hour job costs $y = 65(4) + 95 = 260 + 95 = 355$, so $355

The homeowner can also read the graph: every additional hour shifts the total cost up by $65 (the slope), and even a zero-hour visit starts at $95 (the y-intercept). This structure — a flat fee plus a per-unit rate — appears in utility bills, phone plans, rental agreements, and countless other everyday costs.

Common Mistakes to Avoid

1. Mixing up $m$ and $b$. In $y = 4x + 3$, the slope is $4$ (the coefficient of $x$) and the y-intercept is $3$ (the constant). Do not confuse them.

2. Forgetting the sign of $b$. In $y = 2x - 6$, the y-intercept is $-6$, not $6$. The subtraction sign is part of the value.

3. Not fully isolating $y$. When converting from standard form, you must divide every term by the coefficient of $y$ — not just the $y$-term itself.

4. Treating $y = mx + b$ as the only form. Some problems are easier in point-slope or standard form. Slope-intercept form is ideal for graphing and reading slope/intercept, but it is not always the best starting point for writing an equation.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Slope-intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept
• To identify $m$ and $b$, match the equation to the pattern — the coefficient of $x$ is the slope, the constant is the y-intercept
• To convert from other forms, solve for $y$ by isolating it on one side of the equation
• To graph, plot the y-intercept first, then use the slope to find additional points
• In real-world contexts, the slope is the rate (cost per hour, price per unit) and the y-intercept is the starting value (flat fee, fixed cost)

Return to Algebra 1 for more topics in this section.