Line Equations, Parallelism and Perpendicularity

Line Equations, Parallelism and Perpendicularity

A straight line can be written as $y=mx+c$, where m is gradient and c is y-intercept. Parallel lines have equal gradients; perpendicular non-vertical lines have gradients whose product is -1. To find an intersection, solve the two equations simultaneously and check the resulting coordinate in both lines.

Worked reasoning

For $y=2x+1$ and $y=-x+7$, set equal: $2x+1=-x+7$, so $3x=6$, $x=2$, $y=5$. Intersection is $(2,5)$.

Exam method

1. Put both equations in compatible form. 2. Equate/substitute to find x. 3. Find y, write an ordered pair, and verify in both equations.

Connect algebra and the plane

When solving simultaneous line equations, the algebraic x and y should match the intersection on a rough sketch. If the calculated point is far from where your lines appear to meet, revisit substitution signs. For a line through a point, write the gradient and coordinate before selecting a form. A clean $y=mx+c$ answer makes gradient and intercept readable, while point-gradient working can be useful before expanding. State why lines are parallel or perpendicular using their gradients.

One-minute retrieval: Line Equations, Parallelism and Perpendicularity

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.