Rearranging general line equations
Convert ax + by = c into gradient–intercept form and interpret the exceptional cases.
Section Paper 4 · Extended · Calculator
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Rearrange a linear equation.
- Interpret y = mx + c.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Divide 2y = −3x + 6 by two.
What you will learn
- Find gradient and intercept from ax + by = c.
- Give equivalent simplified line equations.
- Handle equations representing vertical or horizontal lines.
Isolate y before reading coefficients
For 3x + 2y = 6, subtract 3x to get 2y = −3x + 6 and divide every term by two. The result y = −(3/2)x + 3 has gradient −3/2 and y-intercept (0, 3). The coefficient of x in the original form is not automatically the gradient.
In ax + by = c with b ≠ 0, rearrangement gives y = −(a/b)x + c/b. The minus sign comes from moving ax to the other side. This is a relationship between forms, not a reason to ignore the algebra steps. Check a point or intercept in both versions to confirm they describe the same line.
Equivalent forms describe one line
An equation can be multiplied or divided throughout by a non-zero number without changing its solutions. The forms 3x + 2y = 6 and 6x + 4y = 12 therefore describe the same line. Divide by a common factor to give a fully simplified form where appropriate.
To remove fractions in y = (2/3)x − 4, multiply every term by three to obtain 3y = 2x − 12, or 2x − 3y = 12. Keep each sign when moving terms. The form requested by the question determines whether the final answer should isolate y or use integer coefficients.
Do not divide by a coefficient that is zero
If an equation has no y-term, such as 2x = 6, it is the vertical line x = 3. It has undefined gradient and cannot be put into y = mx + c. If there is no x-term, such as −4y = 8, it is the horizontal line y = −2 with gradient zero.
To check intercepts, set the other coordinate to zero in the original equation. For 3x + 2y = 6, the intercepts are (2, 0) and (0, 3). Plotting these two points gives a quick graph, but a third point still helps catch mistakes. The zero equation 0 = 0 does not specify one unique line.
Pause and explain
What is the gradient of 3x + 2y = 6?
Worked example
Find the gradient and y-intercept of 5x − 2y = 8.
Show the worked solution
- Subtract 5x: −2y = 8 − 5x.
- Divide every term by −2: y = (5/2)x − 4.
- The gradient is 5/2 and the y-intercept is (0, −4).
Answer m = 5/2; intercept (0, −4)
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find m and c in 4x + 2y = 10.
Give me a hint
Move the x-term and divide the whole right side by two.
Compare my reasoning
- 2y = −4x + 10.
- y = −2x + 5.
- Hence m = −2 and c = 5; the intercept point is (0, 5).
m = −2, c = 5
Look for these in your work
- Divided every term.
- Distinguished intercept value from point.
Write y = (2/3)x − 4 in a simplified form ax + by = c with integer coefficients.
Give me a hint
Multiply by three before collecting terms.
Compare my reasoning
- 3y = 2x − 12.
- Move terms to give 2x − 3y = 12.
- The coefficients have no common factor greater than one; check x = 0 gives y = −4.
2x − 3y = 12
Look for these in your work
- Removed fractions from every term.
- Preserved the intercept through rearrangement.
Two routes are described by 6x + 4y = 12 and 3x + 2y = 6. Are they different parallel lines or the same line? Explain.
Give me a hint
Compare the entire equations after dividing by a common factor.
Compare my reasoning
- Divide the first equation by two.
- It becomes exactly 3x + 2y = 6.
- Every solution is shared, so the two descriptions represent the same line.
The same line
Look for these in your work
- Compared all coefficients and the constant.
- Distinguished coincident from distinct parallel lines.
Common mistakes
- Reading a as the gradient in ax + by = c.
- Dividing only one term.
- Missing the sign change when dividing by a negative.
What can you explain now?
Find the gradient of 2x − 5y = 10.
Compare with the explanation
2/5
Rearranging gives −5y = 10 − 2x, then y = (2/5)x − 2.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Rearrange one line into two different forms tomorrow and verify both at an intercept and another point.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.