Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — rearranging before reading the gradient
Find the gradient and the y-intercept of the line 2y=6x+8.
- 1The equation is not yet in the form y=mx+c.Nothing can be read off until the left side is a single y.
- 2Divide every term by 2: y=3x+4Every term — the 6x and the 8, not just one of them.
- 3Gradient m=3The number multiplying x.
- 4y-intercept (0,4)The number on its own. Answering '6' from the original equation is the standard trap here.
Example 2 — the equation of a line through two points
Find the equation of the straight line passing through (1,4) and (3,10).
- 1m=3−110−4=26=3Gradient first. Subtract in the same order top and bottom.
- 2So the line is y=3x+c.Put the gradient in and leave c unknown.
- 3Substitute (1,4): 4=3(1)+cEither point works — pick whichever has the friendlier numbers.
- 4c=1, so y=3x+1
- 5Check with the other point: 3(3)+1=10 ✓Use the point you did not substitute. This catches a slip in the gradient.
Example 3 — drawing a line from its intercepts
Find where y=2x−6 crosses each axis, and use those points to draw it.
- 1For the y-intercept, set x=0: y=−6Any point on the y-axis has x=0.
- 2So the line passes through (0,−6).
- 3For the x-intercept, set y=0: 0=2x−6Any point on the x-axis has y=0.
- 42x=6, so x=3, giving (3,0).
- 5Plot (0,−6) and (3,0) and rule a line through them.Two points fix a line, but plot a third — say x=1, y=−4 — as a check.