Equations of straight linesEdexcel A-Level Maths: Revision notes

The gradient of the line through $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2-y_1}{x_2-x_1}.$ Three standard forms are used:

• $y=mx+c$, with gradient $m$ and $y$-intercept $c$.
• $y-y_1=m(x-x_1)$, which gives the line with gradient $m$ through $(x_1,y_1)$.
• $ax+by+c=0$, often with integer $a$, $b$, $c$. Its gradient is $-\frac ab$. Example: through $A(1,4)$ and $B(5,12)$: $m=\frac{12-4}{5-1}=2$, and $y-4=2(x-1)$ gives $y=2x+2$. For $3x-4y+12=0$: $y=\frac34x+3$, so the gradient is $\frac34$.

• Parallel lines have equal gradients: $m_1=m_2$.
• Perpendicular lines satisfy $m_1m_2=-1$, so $m_2=-\frac{1}{m_1}$. Example: the line through $C(6,0)$ perpendicular to $l_1$ (gradient $2$) has gradient $-\frac12$: $y-0=-\frac12(x-6)$, so $x+2y-6=0$. The line $4x+3y-7=0$ has gradient $-\frac43$ and is perpendicular to $3x-4y+12=0$, whose gradient is $\frac34$, because $\frac34\times\left(-\frac43\right)=-1$.

• Set $x=0$ for the $y$-intercept and $y=0$ for the $x$-intercept. For $3x-4y+12=0$ these are $(0,3)$ and $(-4,0)$, giving a triangle with the origin of area $\frac12\times4\times3=6$.
• Distance between $(x_1,y_1)$ and $(x_2,y_2)$ is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.
• To prove perpendicularity, show $m_1m_2=-1$: for $P(-2,1)$, $Q(4,5)$, $R(6,2)$, $m_{PQ}=\frac23$ and $m_{QR}=-\frac32$.
• Then the area of the right-angled triangle is $\frac12\times\sqrt{52}\times\sqrt{13}=13$.

Many real situations are modelled by a straight line, $C=md+c$. The gradient is the rate of change (cost per mile) and the intercept is the starting value (fixed fee). Example: a $4$-mile journey costs $\pounds11$ and a $10$-mile journey $\pounds20$. $m=\frac{20-11}{10-4}=1.5$, and $C=1.5d+5$. A rival with $R=1.2d+8$ charges the same where $1.5d+5=1.2d+8$, i.e. $d=10$. Other standard models are converting temperatures ($F=1.8C+32$) and distance against time (gradient is speed). A limitation is that a straight line assumes a constant rate, and real situations may have minimum charges, waiting time or limits on the range of values.