Part 12 of 36 · LinesSee the whole course
Standard form of a line: Ax + By = C
Standard form writes a line as Ax + By = C, where A, B and C are whole numbers and A is not negative. Its strength is speed: set y = 0 to get the x-intercept, x = 0 to get the y-intercept, and join the two points. The slope is −A ÷ B.
What standard form looks like
Here are three lines in standard form: 3x + 2y = 12, 2x − 5y = 10 and x + y = 7. Both letters are on the left, and a plain number is on the right. Compare slope-intercept form, y = mx + b, where y is alone.
The usual rules for the form are small ones:
- A, B and C are whole numbers, with no fractions or decimals.
- A is not negative. If it is, multiply everything by −1.
- A, B and C have no common factor. 4x + 6y = 12 is written 2x + 3y = 6.
Graph it with the two intercepts
The points where a line crosses the axes are easy to find in this form, because one of the letters drops out. (More on them in x-intercepts and y-intercepts.)
- x-intercept: set y = 0, solve for x. Ax = C, so x = C ÷ A.
- y-intercept: set x = 0, solve for y. By = C, so y = C ÷ B.
- Plot both points and draw the line through them.
Try it. Change A, B and C and watch the two dots move.
- y = 0: 3x = 12, so x = 4
- x = 0: 2y = 12, so y = 6
- y = (−3/2)x + 6
Example 1: 3x + 2y = 12
Set y = 0: 3x = 12, so x = 4. The line crosses the x-axis at (4, 0).
Set x = 0: 2y = 12, so y = 6. It crosses the y-axis at (0, 6).
Join the two dots. Check a third point: x = 2 gives 6 + 2y = 12, so y = 3, and (2, 3) is on the line ✓
Example 2: with a negative, 2x − 5y = 10
Set y = 0: 2x = 10, so x = 5. The point is (5, 0).
Set x = 0: −5y = 10, so y = −2. The point is (0, −2).
The line falls to the left and rises to the right, so the slope is positive: −A ÷ B = −2 ÷ (−5) = 2/5.
Standard form to slope-intercept form
To get y = mx + b, solve for y. For 3x + 2y = 12:
Subtract 3x from both sides: 2y = −3x + 12
Divide everything by 2: y = −(3/2)x + 6
So m = −3/2 and b = 6. This matches the shortcuts: m = −A ÷ B = −3/2 and b = C ÷ B = 12 ÷ 2 = 6.
Remember to divide every term by 2, including the 12 on the right.
Slope-intercept form to standard form
Go the other way by clearing the fractions and moving the x-term across. Take y = (2/3)x − 4:
Multiply every term by 3: 3y = 2x − 12
Subtract 2x from both sides: −2x + 3y = −12
A must not be negative, so multiply everything by −1: 2x − 3y = 12
Two special cases
If A = 0, the x-term vanishes. 2y = 6 is the horizontal line y = 3. If B = 0, the y-term vanishes. 3x = 6 is the vertical line x = 2. Both are covered in horizontal and vertical lines.
Watch it explained in under a minute
Read the video transcript
Here is a line: 3x + 2y = 12. This is standard form. Ax + By = C.
To find where it crosses the x-axis, set y = 0. 3x = 12, so x = 4.
Now the y-axis. Set x = 0. 2y = 12, so y = 6.
Join the two dots. That is the whole line.
One more shortcut. The slope is −A over B. Here, −3 over 2.
Want practice questions and more examples? Full guide: pipmath.app/blog
The five mistakes to avoid
- Swapping the intercepts. Setting x = 0 gives the y-intercept. Setting y = 0 gives the x-intercept.
- Dropping the minus sign in the slope. It is −A ÷ B, not A ÷ B.
- Moving a term without changing its sign. 3x + 2y = 12 becomes 2y = −3x + 12, not 2y = 3x + 12.
- Dividing only some of the terms when you solve for y. Divide every term by B.
- Leaving fractions in standard form. Multiply through to clear them, so A, B and C are whole numbers.
Try it yourself
Tap a question to see the answer.
1. Find both intercepts of 4x + 3y = 12
Set y = 0: 4x = 12, so x = 3, and the point is (3, 0). Set x = 0: 3y = 12, so y = 4, and the point is (0, 4).
2. Write y = −2x + 5 in standard form
Add 2x to both sides: 2x + y = 5. A = 2 is positive and there are no fractions, so 2x + y = 5 is standard form.
3. What is the slope of 5x − 2y = 10?
Solve for y: −2y = −5x + 10, so y = (5/2)x − 5. The slope is 5/2. The shortcut gives the same: −A ÷ B = −5 ÷ (−2) = 5/2.
Where this fits in
Standard form is the favorite form for pairs of equations, because the x-terms and y-terms line up in columns. You will use that in solving a system by elimination. Next in this course: how to tell whether a function is linear.