--- title: "Forms of the Equation of a Straight Line" description: "Move between y = mx + c, y - y₁ = m(x - x₁) and ax + by + c = 0, and produce the equation of a line from a point and a gradient or from two points." canonical: https://lightmysky.com/learn/mathematics/forms-of-the-equation-of-a-straight-line-mt_dhIIxdFwcp source: https://lightmysky.com/learn/mathematics/forms-of-the-equation-of-a-straight-line-mt_dhIIxdFwcp.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Forms of the Equation of a Straight Line Move between y = mx + c, y - y₁ = m(x - x₁) and ax + by + c = 0, and produce the equation of a line from a point and a gradient or from two points. Subject: Mathematics · Area: Geometry · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/forms-of-the-equation-of-a-straight-line-mt_dhIIxdFwcp ## Ready when they can - Find the equation of the line through (2, 5) and (6, 13) - Rewrite 3x + 4y - 12 = 0 in gradient-intercept form and state its gradient - Find where a given line meets each axis ## Lesson: Forms of the Equation of a Straight Line Last stop you turned a rule into a picture. Now the picture comes first. On the Ridgeway plan the practice strip is a straight fence line, and the crew has marked two pegs on the squared paper. They need the exact equation, not a drawing. A line's equation is a test: every point on the line makes it true, and every point off the line does not. That is why setting y = 0 finds where the line meets the x-axis, and setting x = 0 finds where it meets the y-axis. Those two crossings are often all a plan needs. You already write a line as y = mx + c, gradient m and height c at x = 0. Two more forms earn their place. The point-gradient form y - y₁ = m(x - x₁) takes a gradient and one point (x₁, y₁), no rearranging needed. Some books say slope for gradient, so this is also point-slope form. The standard form ax + by + c = 0 clears fractions and keeps whole numbers. Only one line sits outside y = mx + c: the vertical line x = k, whose gradient is undefined. A horizontal line y = k is that form with m = 0. *(drawing: Same line, three costumes. Pick the one that fits what you were given.)* **Example.** Write the line through (5, 3) with gradient 4. Start with the form that takes those two pieces straight: y - y₁ = m(x - x₁), so y - 3 = 4(x - 5). That is already an answer. To reach gradient-intercept form, expand and tidy: y - 3 = 4x - 20, so y = 4x - 17. For standard form, move everything to one side: 4x - y - 17 = 0. Check the point in that last one: 4(5) - 3 - 17 = 0. All three describe the same line, and which you write depends on what the question wants next. **Example.** Now the line through the two pegs at (2, 5) and (6, 13), with no gradient given. Find it first: divide the change in y by the change in x, so m = (13 - 5) / (6 - 2) = 8 / 4 = 2. Then use either point in the point-gradient form. Taking (2, 5) gives y - 5 = 2(x - 2), which tidies to y = 2x + 1. Taking (6, 13) gives the same line. Standard form is 2x - y + 1 = 0, and a fractional gradient would need every term multiplied through first. *(drawing: The gradient step is the only extra one. Keep the points in the same order top and bottom.)* **Example.** Standard form hides the gradient, so dig it out. Take 3x + 4y - 12 = 0. Move the x term and the number across: 4y = -3x + 12. Divide every term by 4: y = -(3/4)x + 3. So the gradient is -3/4 and the line cuts the y-axis at (0, 3). For the other crossing set y = 0 in the original: 3x - 12 = 0, so x = 4 and the line cuts the x-axis at (4, 0). Going the other way, multiply through by the denominator until every coefficient is a whole number. **Recap.** One line, three ways to write it. Use y = mx + c to read a gradient and a y-intercept, y - y₁ = m(x - x₁) when you have a point and a gradient, and ax + by + c = 0 to keep whole numbers. From two points, work out the gradient first and then use either point. Set y = 0 for the x-axis crossing and x = 0 for the y-axis crossing. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Sketching a Curve from Its Factorised Form](https://lightmysky.com/learn/mathematics/sketching-a-curve-from-its-factorised-form-mt_x4tZLdH8fh) - [Plotting Linear Graphs](https://lightmysky.com/learn/mathematics/plotting-linear-graphs-mt_-3udyo6VyB) ## Opens up - [Parallel and Perpendicular Lines in Coordinates](https://lightmysky.com/learn/mathematics/parallel-and-perpendicular-lines-in-coordinates-mt_xNEmm139Ka)