# Slope-intercept form and how to write the equation of a line Canonical: https://duckyhelper.com/learn/algebra-1/slope-intercept-form/ Updated: 2026-10-01 Slope-intercept form is \(y = mx + b\). The number \(m\) is the slope and \(b\) is the y-intercept, where the line crosses the y-axis. To write the equation of a line, find the slope first, then plug in one point to find \(b\). Point-slope form, \(y - y_1 = m(x - x_1)\), is a faster start when you know a point. Standard form is \(Ax + By = C\). ## The key idea $$ y = mx + b $$ Read it as: start at \(b\) on the y-axis, then for every 1 step to the right, go up \(m\). In \(y = -2x + 3\), the line crosses the y-axis at \((0, 3)\) and drops 2 for every step right, so it also passes through \((1, 1)\) and \((2, -1)\). **Three ways to write the same line** | Form | Equation | Best when | | --- | --- | --- | | Slope-intercept | \(y = mx + b\) | you want to graph it or read the slope and y-intercept | | Point-slope | \(y - y_1 = m(x - x_1)\) | you know a slope and any point | | Standard | \(Ax + By = C\) | the answer must have whole numbers, or you want both intercepts fast | All three describe the same line. You can move between them with ordinary algebra. ## Worked examples **Example 1: a slope and a point** Problem: Write the equation of the line with slope 4 that passes through \((2, 5)\). 1. Start with \(y = 4x + b\). Put in the point: \(x = 2\) and \(y = 5\). $$ 5 = 4(2) + b $$ 2. Multiply. $$ 5 = 8 + b $$ 3. Subtract 8. $$ b = -3 $$ 4. Write the full equation and check the point: \(4(2) - 3 = 5\). It works. Answer: \(y = 4x - 3\) **Example 2: two points** Problem: Write the equation of the line through \((-2, 7)\) and \((4, -5)\). 1. Find the slope. $$ m = \frac{-5 - 7}{4 - (-2)} = \frac{-12}{6} = -2 $$ 2. Use either point to find b. Here we use \((-2, 7)\). $$ 7 = -2(-2) + b $$ 3. Simplify: \(7 = 4 + b\), so subtract 4. $$ b = 3 $$ 4. Check the other point: \(-2(4) + 3 = -5\). It works. Answer: \(y = -2x + 3\) **Example 3: point-slope form, then standard form** Problem: Write the equation of the line through \((3, -1)\) that is parallel to \(y = \frac{1}{2}x + 6\). Give it in slope-intercept form and in standard form with whole numbers. 1. Parallel lines have the same slope, so \(m = \frac{1}{2}\). Plug the point into point-slope form. Since \(y_1 = -1\), \(y - (-1)\) becomes \(y + 1\). $$ y + 1 = \frac{1}{2}(x - 3) $$ 2. Distribute the \(\frac{1}{2}\) to both terms. $$ y + 1 = \frac{1}{2}x - \frac{3}{2} $$ 3. Subtract 1, which is \(\frac{2}{2}\). $$ y = \frac{1}{2}x - \frac{5}{2} $$ 4. For standard form, multiply every term by 2 to clear the fractions. $$ 2y = x - 5 $$ 5. Move the x term to the left side. Standard form usually keeps A positive. $$ x - 2y = 5 $$ Answer: \(y = \frac{1}{2}x - \frac{5}{2}\), which is \(x - 2y = 5\) in standard form. **Example 4 (test-hard): perpendicular line and its x-intercept** Problem: A line passes through \((4, 1)\) and is perpendicular to \(2x + 3y = 6\). Where does it cross the x-axis? 1. The given line has slope \(-\frac{A}{B} = -\frac{2}{3}\). The perpendicular slope is the negative reciprocal, \(\frac{3}{2}\). Use point-slope form. $$ y - 1 = \frac{3}{2}(x - 4) $$ 2. Distribute and add 1. $$ y = \frac{3}{2}x - 5 $$ 3. The x-intercept is where \(y = 0\). $$ 0 = \frac{3}{2}x - 5 $$ 4. Add 5, then multiply by \(\frac{2}{3}\). $$ x = \frac{10}{3} $$ Answer: \(\left(\frac{10}{3}, 0\right)\) ## Common mistakes - **Mixing up b and the x-intercept.** \(b\) is where the line crosses the **y**-axis, the value of y when \(x = 0\). - **Sign slips in point-slope form.** For the point \((3, -1)\), you get \(y + 1 = m(x - 3)\). Fix: write \(y - (-1)\) first, then simplify. - **Distributing to only one term.** \(\frac{1}{2}(x - 3)\) is \(\frac{1}{2}x - \frac{3}{2}\), not \(\frac{1}{2}x - 3\). - **Reading the slope straight from standard form.** In \(2x + 3y = 6\), the slope is \(-\frac{2}{3}\), not 2. - **Swapping x and y when you plug in a point.** In \((2, 5)\), 2 is x and 5 is y. Fix: write \(x = 2,\ y = 5\) before substituting. ## Quick methods > **Tip: Graph standard form with the cover-up method** > > For \(2x + 3y = 6\): cover the x term to get \(3y = 6\), so the y-intercept is \((0, 2)\). Cover the y term to get \(2x = 6\), so the x-intercept is \((3, 0)\). Plot both and connect them. This works because each intercept has one coordinate equal to 0. > **Note: Check with both points** > > When you write a line through two points, plug in the point you did not use to find b. If it fits, your equation is right. A graphing calculator like Desmos does the same check visually: type the points and your equation and see if the line hits both. ## Practice **5 practice questions** 1. What are the slope and y-intercept of \(y = 7 - 3x\)? A. slope \(7\), y-intercept \(-3\) B. slope \(-3\), y-intercept \(7\) C. slope \(3\), y-intercept \(7\) D. slope \(-3\), y-intercept \(-7\) Answer: slope \(-3\), y-intercept \(7\). Rewrite it as \(y = -3x + 7\). The slope is the number multiplied by x, sign included, so it is \(-3\). The order of the terms does not change which number is which. 2. Which equation describes the line through \((0, -4)\) and \((2, 2)\)? A. \(y = 3x - 4\) B. \(y = -4x + 3\) C. \(y = \frac{1}{3}x - 4\) D. \(y = 3x + 4\) Answer: \(y = 3x - 4\). Slope: \(\frac{2 - (-4)}{2 - 0} = \frac{6}{2} = 3\). The point \((0, -4)\) is on the y-axis, so \(b = -4\). The choice \(\frac{1}{3}x\) uses run over rise, and \(-4x + 3\) swaps m and b. 3. Which is an equation of the line with slope \(-2\) through \((-1, 5)\)? A. \(y - 5 = -2(x + 1)\) B. \(y + 5 = -2(x - 1)\) C. \(y - 5 = -2(x - 1)\) D. \(y - 1 = -2(x + 5)\) Answer: \(y - 5 = -2(x + 1)\). Point-slope form is \(y - y_1 = m(x - x_1)\). With \(x_1 = -1\), \(x - (-1)\) becomes \(x + 1\). The other choices flip a sign or swap the coordinates. 4. Which is \(y = \frac{2}{3}x - 4\) written in standard form with whole numbers? A. \(2x - 3y = 12\) B. \(2x + 3y = 12\) C. \(2x - 3y = 4\) D. \(3x - 2y = 12\) Answer: \(2x - 3y = 12\). Multiply every term by 3: \(3y = 2x - 12\). Subtract \(2x\) and multiply by \(-1\) so the x term is positive: \(2x - 3y = 12\). The choice ending in 4 forgot to multiply the \(-4\) by 3. 5. Which line is parallel to \(6x - 2y = 8\) and passes through \((1, 1)\)? A. \(y = 3x - 2\) B. \(y = -3x + 4\) C. \(y = 3x - 4\) D. \(y = -\frac{1}{3}x + \frac{4}{3}\) Answer: \(y = 3x - 2\). Solve for y: \(y = 3x - 4\), so the slope is 3. That is the original line itself, and it misses \((1, 1)\). Use point-slope: \(y - 1 = 3(x - 1)\), so \(y = 3x - 2\). The last choice is the perpendicular line through \((1, 1)\). ## Frequently asked questions ### What do m and b stand for in y = mx + b? \(m\) is the slope, how much y changes when x goes up by 1. \(b\) is the y-intercept, the value of y when x is 0, which is where the line crosses the y-axis. The letters are just tradition; what matters is their position in the equation. ### How do I find b if I know the slope and a point? Put the slope and the point's x and y into \(y = mx + b\). Everything is a number except b, so solve that small equation. For slope 4 through \((2, 5)\): \(5 = 8 + b\), so \(b = -3\). ### When should I use point-slope form? Use it when you know a slope and a point that is not the y-intercept. You can write the equation right away with no solving. If your teacher wants slope-intercept form, distribute and solve for y afterward. ### How do I change standard form to slope-intercept form? Solve for y. Move the x term to the right side, then divide every term by the number in front of y. For \(2x + 3y = 6\): \(3y = -2x + 6\), so \(y = -\frac{2}{3}x + 2\). ## Related - [How to find the slope of a line](https://duckyhelper.com/learn/algebra-1/slope/) - [How to solve systems of equations](https://duckyhelper.com/learn/algebra-1/systems-of-equations/) - [Linear functions and slope on the SAT](https://duckyhelper.com/learn/sat-math/linear-functions/) - [Algebra 1 study guides](https://duckyhelper.com/learn/algebra-1/) ## Try asking Ducky - "How do I write the equation of the line on this graph?" - "I got y = 4x + 13 for this one. Where did I go wrong finding b?" - "Can you show me how to graph 2x + 3y = 6 step by step?" ## Get DuckyHelper Free to start. The web app works in any browser, Chromebooks included; the Mac app can also draw on your real screen. [Try it free in your browser](https://app.duckyhelper.com/?utm_source=duckyhelper.com&utm_medium=learn) or [Get the Mac app](https://duckyhelper.com/download/)