Quick questions on Coordinate geometry of the straight line explained: O-Level E-Maths

A straight line has equation $y = mx + c$, with gradient $m$ and $y$-intercept $c$. To determine it you need either two points, or one point and the gradient. From two points, compute the gradient, then substitute one point to find $c$.

Two lines are parallel exactly when their gradients are equal. So a line parallel to $y = 3x - 1$ has gradient $3$, differing only in its intercept. Comparing gradients is the test for parallelism.

Two lines are perpendicular when the product of their gradients is $-1$:

A natural follow-on is finding the point where two lines cross, which you do by solving their equations simultaneously. Set the two expressions for $y$ equal (or use elimination), solve for $x$, then substitute back to get $y$. The intersection of $y = 2x + 1$ and $y = -x + 7$ comes from $2x + 1 = -x + 7$, so $3x = 6$, $x = 2$, and $y = 5$, giving the point $(2, 5)$. This single skill underlies many coordinate-geometry tasks, such as finding the foot of a perpendicular or the vertex of a shape, because those points are always intersections of two lines you can write down.

E-Maths sometimes gives a line as $ax + by = c$ rather than $y = mx + c$, and you cannot read the gradient off it directly. Rearrange to make $y$ the subject first: from $3x + 2y = 12$, you get $y = -\tfrac{3}{2}x + 6$, so the gradient is $-\tfrac{3}{2}$. Only after this rearrangement can you apply the parallel or perpendicular tests. Converting any line into gradient-intercept form before comparing gradients is a small but essential habit, since a sign error during rearrangement would flip every later conclusion about parallel or perpendicular.

If the answer must be $y = mx + c$, finish the rearrangement.

State the gradient of a line perpendicular to $y = 4x - 3$. [1 mark]

Find the equation of the line with gradient $-3$ through $(2, 1)$. [2 marks]

Are the lines $y = 2x + 1$ and $y = 2x - 5$ parallel, perpendicular or neither? [1 mark]