COORDINATE GEOMETRY · HL

Equation of a Line

Finding the equation of a line from a point and slope, or from two points.

Section 1 of 5

Building an equation

I have €10 and will save €5 every day. Form a table for savings. Show on a graph and form an equation.

Change $=$ variables. Money changes → dependent $= y$. Time changes → independent $= x$.

Time	Money	Point
0	10	$(0,10)$
1	15	$(1,15)$
2	20	$(2,20)$
3	25	$(3,25)$
4	30	$(4,30)$
$x$	$y = 5x + 10$

$y = 5x + 10 \qquad x = \text{time}, \quad y = \text{money}$

After 20 days

$y = 5(20) + 10 = 110$

Before €1000

$5x + 10 = 1000$

$5x = 990$

$x = \dfrac{990}{5}$

Section 2 of 5

The formula: point + slope

$y - y_1 = m(x - x_1)$

What you need

1.Point $(x_1, y_1)$

2.Slope $m$

3.Never changes in $y$ and $x$.

(i) Line through $(2,20)$ with slope $5$

$(2,20) \to x_1,y_1 \qquad m = 5$

$y - 20 = 5(x - 2)$

$y - 20 = 5x - 10$

$y = 5x + 10$

(ii) Line through $(2,50)$ with slope $-3$

$(2,50) \to x_1,y_1 \qquad m = -3$

$y - 50 = -3(x - 2)$

$y - 50 = -3x + 6$

$y = -3x + 56$

$3x + y = 56$

Section 3 of 5

More point & slope

(i) Through $(-1,6)$ with slope $-8$

$y - 6 = -8(x + 1)$

$y - 6 = -8x - 8$

$8x + y = -8 + 6$

$8x + y = -2$

(ii) Point $(-1,7)$, slope $-6$

$y - 7 = -6(x + 1)$

$y - 7 = -6x - 6$

$6x + y = -6 + 7$

$6x + y = 1$

(iii) $(-3,7)$, $m = -2$

$y - 7 = -2(x + 3)$

$y - 7 = -2x - 6$

$2x + y = -6 + 7$

$2x + y = 1$

(iv) $(2,7)$, $m = 4$

$y - 7 = 4(x - 2)$

$y - 7 = 4x - 8$

$-4x + y = -8 + 7$

$4x - y = 1$

(v) $(2,-9)$, slope $6$

$y + 9 = 6(x - 2)$

$y + 9 = 6x - 12$

$-6x + y = -12 - 9$

$-6x + y = -21$

$6x - y = 21$

(vi) Through $(-3,1)$, slope $-5$

$y - 1 = -5(x + 3)$

$y - 1 = -5x - 15$

$5x + y = -15 + 1$

$5x + y = -14$

(vii) Point $(-1,4)$, slope $3$

$y - 4 = 3(x + 1)$

$y - 4 = 3x + 3$

$-3x + y = 3 + 4$

$-3x + y = 7$

$3x - y = -7$

General form: $ax + by = c$.

Section 4 of 5

Fraction slopes — clear the fraction

(i) Point $(-1,5)$, $m = -\dfrac{2}{3}$

$3(y - 5) = 3\left(-\dfrac{2}{3}(x + 1)\right)$

$3y - 15 = -2x - 2$

$2x + 3y = -2 + 15$

$2x + 3y = 13$

(ii) $(-4,1)$, $m = -\dfrac{3}{5}$

$5(y - 1) = 5\left(-\dfrac{3}{5}(x + 4)\right)$

$5y - 5 = -3x - 12$

$3x + 5y = -12 + 5$

$3x + 5y = -7$

(iii) $(-2,-1)$, $m = \dfrac{2}{7}$

$7(y + 1) = 7\left(\dfrac{2}{7}(x + 2)\right)$

$7y + 7 = 2x + 4$

$-2x + 7y = 4 - 7$

$-2x + 7y = -3$

$2x - 7y = 3$

(iv) $(3,5)$, $m = \dfrac{4}{5}$

$5(y - 5) = 5\left(\dfrac{4}{5}(x - 3)\right)$

$5y - 25 = 4x - 12$

$-4x + 5y = -12 + 25$

$-4x + 5y = 13$

$4x - 5y = -13$

Section 5 of 5

Line through two points

First find the slope, then use point & slope.

$m = \dfrac{y_2 - y_1}{x_2 - x_1} \qquad y - y_1 = m(x - x_1)$

(i) Through $(-1,3)$ and $(2,5)$

$m = \dfrac{5-3}{2+1} = \dfrac{2}{3}$

$3(y - 3) = 3\left(\dfrac{2}{3}(x + 1)\right)$

$3y - 9 = 2x + 2$

$-2x + 3y = 11$

$2x - 3y = -11$

(ii) $(3,-5)$ $(-2,1)$

$m = \dfrac{1+5}{-2-3} = \dfrac{6}{-5}$

$5(y + 5) = 5\left(-\dfrac{6}{5}(x - 3)\right)$

$5y + 25 = -6x + 18$

$6x + 5y = 18 - 25$

$6x + 5y = -7$

(iii) $(-4,3)$ $(-1,-6)$

$m = \dfrac{-6-3}{-1+4} = \dfrac{-9}{3} = -3$

$y - 3 = -3(x + 4)$

$y - 3 = -3x - 12$

$3x + y = -12 + 3$

$3x + y = -9$

(iv) $(-2,1)$ $(3,6)$

$m = \dfrac{6-1}{3+2} = 1$

$y - 1 = 1(x - 2)$

$y - 1 = 1x - 2$

$-x + y = -2 + 1$

$x - y = 1$

(v) $A(-3,2)$ $B(-1,4)$ — equation of line through $AB$

$m = \dfrac{4-2}{-1+3} = \dfrac{2}{2} = 1$

$y - 2 = 1(x + 3)$

$y - 2 = x + 3$

$-x + y = 5$

$x - y = -5$

SUM

The lot in one box

Equation of a line toolkit

1.$y - y_1 = m(x - x_1)$ — need a point and a slope.

2.Two points first: $m = \dfrac{y_2-y_1}{x_2-x_1}$, then point & slope.

3.Fraction slope: multiply both sides to clear it.

4.Tidy to general form $ax + by = c$.

End of lesson

Equation of a Line — HL · Mathslive.ie