KJSEA 2025 Maths Q28 — Equation of a Line and x-Intercept

KJSEA 2025 Grade 9 Geometry

Published

The Question

“One side of a parallelogram passes through the point (1, 4) and has a gradient of 1/2. Part A: find the equation of the line. Part B: find the x-intercept of the line.”

1

Choose the point-gradient form

Because you already know one point on the line and its gradient, the quickest route is the point-gradient form of a straight line. You substitute the known point in for the reference coordinates and the given gradient in for m, leaving x and y as the general variables.

y−y1=m(x−x1)y - y_{1} = m(x - x_{1})
2

Substitute the point and gradient

Put the coordinates of the given point in place of x1 and y1, and put the gradient in place of m. Here the point is (1, 4) and the gradient is one half.

y−4=12(x−1)y - 4 = \frac{1}{2}(x - 1)
3

Expand and tidy up

Multiply out the bracket on the right, then add 4 to both sides to make y the subject. This gives the line in the neat slope-intercept form, which is also the answer to Part A.

y−4=12x−12y - 4 = \frac{1}{2}x - \frac{1}{2}
y=12x−12+4y = \frac{1}{2}x - \frac{1}{2} + 4
y=12x+72y = \frac{1}{2}x + \frac{7}{2}
4

Set y to zero for the x-intercept

The x-intercept is the point where the line crosses the x-axis, and every point on the x-axis has a y-coordinate of zero. So substitute y equals 0 into the equation and solve for x.

0=12x+720 = \frac{1}{2}x + \frac{7}{2}
12x=−72\frac{1}{2}x = -\frac{7}{2}
x=−7x = -7
5

State the intercept as a point

Since the intercept lies on the x-axis, its y-coordinate is 0. Pair the value of x you found with a y of 0 to write the crossing point in full.

(−7, 0)(-7,\ 0)

Final Result

Part A: the equation of the line is y = (1/2)x + 7/2. Part B: the line crosses the x-axis at the point (-7, 0).

Why this method works

The point-gradient form works because a straight line has a single, constant gradient, so the rise over the run between any general point (x, y) and the known point (1, 4) must always equal the gradient. Writing that ratio equal to one half and clearing the fraction reproduces the line exactly. The x-intercept step works because the x-axis is defined by y = 0, so forcing y to zero pinpoints exactly where the line meets that axis.

✓

Substitute x = -7 back in: y = (1/2)(-7) + 7/2 = -7/2 + 7/2 = 0, confirming (-7, 0) lies on the line.