What is slope-intercept form?

Slope-intercept form is a standard way to write the equation of a straight line: y = mx + b. It’s popular because you can “read” the graph instantly. The number m tells you the line’s tilt (slope), and the number b tells you where the line hits the y-axis (the y-intercept).

Think of m as the line’s speed: it’s the amount y changes when x goes up by 1. If m = 3, then every time x increases by 1, y increases by 3 — the line rises quickly. If m = -1/2, then every time x increases by 2, y decreases by 1 — the line falls gently. If m = 0, the line is perfectly flat (horizontal).

The intercept b is simpler: it’s the y-value when x is 0. Plug in x = 0 and you get y = b. So if the equation is y = 2x + 5, the line crosses the y-axis at (0, 5). This is why slope-intercept form is so useful for fast graphing: start at the intercept, then use the slope to move “rise over run” from that starting point.

• It’s easy to graph: start at (0, b), then apply rise/run using m.
• It shows how y depends on x: every line is “multiply x by m, then shift by b”.
• It connects to real life: m is rate (dollars per hour, miles per minute), and b is a starting amount (base fee).

The key formulas you’ll use

This calculator supports three common input styles. Under the hood, everything becomes y = mx + b.

• Equation: y = mx + b
• Y-intercept: (0, b)
• X-intercept (if m ≠ 0): set y = 0 ⇒ x = -b/m

• Slope: m = (y₂ − y₁) / (x₂ − x₁)
• Then find b using one point: b = y₁ − m·x₁

• Point-slope form: y − y₁ = m(x − x₁)
• Convert to slope-intercept by distributing and isolating y.
• Shortcut: b = y₁ − m·x₁

Notice the same shortcut appears in multiple methods: once you know a slope and a point on the line, you can always compute b using b = y − mx.

How to convert two points into y = mx + b (the classic method)

Two-point problems are the most common in algebra and coordinate geometry. The strategy is always the same: (1) find the slope, (2) use the slope with a point to find b.

• Step 1: slope. m = (11 − 3) / (5 − 1) = 8 / 4 = 2
• Step 2: find b. Use b = y₁ − m·x₁ with (1, 3): b = 3 − 2·1 = 1
• Step 3: write the equation. y = 2x + 1

That’s it. If you want a fast self-check, plug in the second point: x = 5 → y should be 11. 2·5 + 1 = 11. Checks out.

If both x-values are the same, the denominator in the slope formula is 0, which means the line is vertical. Vertical lines cannot be written in slope-intercept form because they are not functions of x. Their equation is x = constant, like x = 4. This calculator will warn you if you enter a vertical line.

How to “read” a line once you have y = mx + b

Once your equation is in slope-intercept form, you can interpret it like a mini story: “Start at b, then move with slope m.”

• m > 0: line rises as you move right (increasing).
• m < 0: line falls as you move right (decreasing).
• m = 0: horizontal line (constant y).
• |m| large: steep line. |m| small: gentle line.

• b: y-value when x = 0 (starting value).
• x-intercept: where the line crosses the x-axis; set y = 0 and solve.

A helpful real-life translation: if y is “total cost” and x is “hours,” then m is cost per hour, and b is the base fee. The equation y = 30x + 15 reads as “$15 starting fee, then $30 per hour.”

From point-slope form to slope-intercept form

Point-slope form is often given as y − y₁ = m(x − x₁). To convert it into slope-intercept form, distribute m and then isolate y.

• Distribute: y − 2 = 3x − 12
• Add 2 to both sides: y = 3x − 10
• So m = 3, b = −10.

If you want a shortcut, skip the distribution and use b = y₁ − m·x₁. With point (4, 2) and m = 3: b = 2 − 3·4 = 2 − 12 = −10. Same answer, faster.

Frequently Asked Questions

• What is slope-intercept form used for?

  It’s used to graph lines quickly, compare rates of change, and interpret linear relationships. It’s one of the most common forms in Algebra 1 and coordinate geometry.

• Can every line be written as y = mx + b?

  Almost — every non-vertical line can. Vertical lines have equations like x = 3 and do not have a slope-intercept form because they fail the “function” rule (one x gives many y values).

• How do I find the x-intercept?

  Set y = 0 and solve: 0 = mx + b ⇒ x = −b/m (as long as m ≠ 0). The intercept point is (−b/m, 0).

• What if my slope is a fraction?

  That’s totally normal. A slope like -3/4 means “down 3, right 4.” When graphing, start at the y-intercept, then apply the rise/run steps repeatedly.

• What if my points have decimals?

  Use them directly. The slope formula works with decimals, fractions, and negatives. If you want a clean final equation, you can convert decimals to fractions (for example, 0.75 = 3/4).

Mini cheat sheet (copy/paste)

• Slope (two points): m = (y₂ − y₁)/(x₂ − x₁)
• Intercept: b = y − mx (use any point on the line)
• Slope-intercept: y = mx + b
• X-intercept: x = −b/m (if m ≠ 0)
• Vertical line: x = constant (no slope-intercept form)

Want to make this page more viral? Screenshot the “cheat sheet” + your equation and post it with your class group chat. (Yes, people actually do this.)

How this tool helps (and how to use it well)

If your teacher asks you to “write the equation of the line in slope-intercept form,” the fastest path is: compute m first, then compute b, then write y = mx + b. This calculator mirrors that exact workflow and prints the steps in plain language so you can learn the pattern — not just get an answer.

For maximum accuracy, always verify with a quick substitution: plug one original point into your final equation. If it works, you’re done. If it doesn’t, the most common problem is a sign mistake in the slope calculation or in b = y − mx.

Disclaimer: This page is for educational support. If you’re stuck, ask your teacher for clarification on the form they want (slope-intercept, point-slope, or standard form).