How to Write Linear Equations? (+FREE Worksheet!)
Writing Linear Equations
Writing a linear equation means turning the facts you’re given, a slope and a point, or two points, into \(y = mx + b\). Find the slope, find the intercept, and you’re done. We’ll cover every starting point, with a solver, practice, and a worksheet maker a tap away.
Write Linear Equations: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Find slopeUse two points, a table, or the coefficient of x in slope-intercept form.
- Find an anchorUse a point or intercept so the line is in the right location.
- Check directionPositive slope rises left to right; negative slope falls left to right.
Worked examples
Find slope from two points
- Change in y is 10 – 4 = 6.
- Change in x is 3 – 1 = 2.
- Divide rise by run.
Write slope-intercept form
- Use y = mx + b.
- Put m = 3 and b = -2.
- Write the line.
Try one before moving on
Write Linear Equations: pop-up practice
Writing a linear equation is the skill of turning whatever you’re given, a slope and a point, or two points, into the equation \(y = mx + b\). It comes up constantly, because once you have the equation you can graph the line, predict values, and compare it to others. The recipe is always the same: find the slope, then find the intercept.
In short: find the slope \(m\), find the y-intercept \(b\) (read it or solve for it), then write \(y = mx + b\). For example, the line through \((0,1)\) and \((2,7)\) is \(y = 3x + 1\).
Slope First, Intercept Second
Every non-vertical line is \(y = mx + b\). The slope \(m\) sets the tilt; the intercept \(b\) sets where it crosses the y-axis. Whatever the problem gives you, your job is to pin down those two numbers.
How to write the equation (3 steps):
- Find the slope (from two points: \(\tfrac{y_2-y_1}{x_2-x_1}\); or it’s given).
- Find \(b\): read it if you have the y-intercept, or plug a point into \(y = mx + b\) and solve.
- Write \(y = mx + b\).
Through \((0,1)\) and \((2,7)\)
Slope \(=\tfrac{7-1}{2-0}=3\); the point \((0,1)\) gives \(b=1\). So \(y = 3x + 1\). The line below climbs 3 for every 1 across, crossing the y-axis at 1.
Write a line’s equationWorked Examples
Slope first, intercept second, each finished line is graphed through its given point.
Example A, Two points (intercept given)
Through \((0,1)\) and \((2,7)\).
- Slope: \(m = \dfrac{7 – 1}{2 – 0} = 3\).
- One point has \(x = 0\), so it’s the intercept: \(b = 1\).
- Write it: \(y = 3x + 1\).
Answer: \(y = 3x + 1\)
Example B, Slope and a point
Slope 2 through \((3,1)\).
- Plug into \(y = mx + b\): \(1 = 2(3) + b\).
- Solve: \(1 = 6 + b\), so \(b = -5\).
- Write it: \(y = 2x – 5\).
Answer: \(y = 2x – 5\)
Example C, Two points (solve for b)
Through \((1,1)\) and \((2,4)\).
- Slope: \(m = \dfrac{4 – 1}{2 – 1} = 3\).
- Use \((1,1)\): \(1 = 3(1) + b\), so \(b = -2\).
- Write it: \(y = 3x – 2\).
Answer: \(y = 3x – 2\)
Example D, A horizontal result
Through \((2,3)\) and \((4,3)\).
- Slope: \(m = \dfrac{3 – 3}{4 – 2} = 0\).
- Zero slope means a flat line at the shared \(y\)-value.
- Write it: \(y = 3\).
Answer: \(y = 3\)
Where You’ll Use It
Any steady real-world relationship becomes a linear equation: a gym’s flat fee plus a per-class rate, a tank draining at a constant speed, a phone plan’s monthly cost. Write the equation once and you can answer “what’s the cost at 30 classes?” or “when will the tank be empty?” without re-reading the data.
Slip-Ups That Cost Easy Points
- Run over rise. Slope is the change in \(y\) over the change in \(x\), y on top.
- Forgetting to solve for \(b\). If the y-intercept isn’t given, plug a point in and solve; don’t guess \(b = 0\).
- Sign mistakes plugging in. \(1 = 2(3) + b\) gives \(b = -5\), not \(5\).
- Order of subtraction. Keep the points in the same order top and bottom of the slope fraction.
Your Turn: Write the Equation
Write each in \(y = mx + b\). Reveal to check.
- Through \((0,2)\) and \((4,10)\)
- Through \((1,1)\) and \((2,4)\)
- Through \((0,-1)\) and \((5,9)\)
- Through \((2,3)\) and \((4,3)\)
Show answers
- \(\color{blue}{y = 2x + 2}\)
- \(\color{blue}{y = 3x – 2}\)
- \(\color{blue}{y = 2x – 1}\)
- \(\color{blue}{y = 3 \text{ (slope } 0)}\)
Make Your Own Worksheet
Generate fresh write-the-equation problems with a full answer key, print or save as a PDF.
Frequently Asked Questions
How do you write linear equations in slope-intercept form?
Find the slope m and the y-intercept b, then substitute them into y = mx + b. If you know the slope and one point, plug them in to solve for b. With slope 4 through (2, 3): 3 = 4(2) + b, so b = −5 and the equation is y = 4x − 5. For a quick check, substituting x = 2 should give y = 3.
How do you write a linear equation given two points?
First find the slope with m = (y₂ − y₁)/(x₂ − x₁), then use either point to find b. For (1, 5) and (4, 11), the slope is (11 − 5)/(4 − 1) = 6/3 = 2. Substituting (1, 5) into y = 2x + b gives 5 = 2 + b, so b = 3. The equation is y = 2x + 3, and (4, 11) checks out: 2(4) + 3 = 11.
How do you write linear equations from a table?
Pick two rows and find the rate of change: the change in y divided by the change in x. Then find the y-value when x = 0, which is the y-intercept. If a table shows x = 0, 1, 2, 3 and y = 7, 10, 13, 16, y increases by 3 each time x increases by 1, so m = 3, and b = 7. The equation is y = 3x + 7.
How do you write linear equations from word problems?
Look for a rate, which becomes the slope, and a starting amount, which becomes the y-intercept. Phrases like per hour or each signal the rate; words like fee, initial or starts with signal the starting value. If a plumber charges a $60 visit fee plus $45 per hour, the cost is C = 45h + 60. For a 3-hour job, C = 45(3) + 60 = $195.
How do you write linear equations in standard form?
Standard form is Ax + By = C, where A, B and C are integers and A is usually positive. Start from any form, move the x and y terms to one side and clear fractions. For y = (2/3)x − 4, multiply every term by 3 to get 3y = 2x − 12, then subtract 2x from both sides: −2x + 3y = −12. Multiply by −1 to get 2x − 3y = 12.
How do you write linear equations in point-slope form?
Point-slope form is y − y₁ = m(x − x₁). Plug in the slope and the coordinates of one point, and you are done; there is no need to find the y-intercept first. For slope −2 through (3, 1), the equation is y − 1 = −2(x − 3). Distributing and adding 1 to both sides turns it into slope-intercept form: y = −2x + 7.
Related Topics
Continue Your Study
Ready for the next step? Pick up right where this lesson leaves off:
Related to This Article
More math articles
- Washington Smarter Balanced Grade 7 Math Worksheets: 95 Free Printable Skill PDFs
- How to Pay for College: Understanding College Payments
- 3rd Grade MAP Math FREE Sample Practice Questions
- The Best Grade 8 ELA Practice Tests for Missouri Students
- Using Strip Diagrams to Represent Fractions
- Using Strip Models to Explain Percent
- Election of 1824 and the Changing Party System: Votes, Rules and Claims
- How to Use Strip Models to Identify Equivalent Expressions?
- 5 Best HiSET Math Study Guides
- Free Grade 3 English Worksheets for Massachusetts Students
What people say about "How to Write Linear Equations"?
No one replied yet.