COORDINATE GEOMETRY · HL

Slope, Drawing & Parallel Lines

Reading slope from an equation, drawing lines, and lines parallel to a given line.

Section 1 of 4

Slope from an equation

Write in the form $y = mx + c$. Then read the slope $m$ and the $y$-intercept $(0, c)$.

(i) Find slope of $2x + y = 3$

$y = -2x + 3$

$m = -2 \qquad (0,3)$

(ii) $3x + y = 8$

$y = -3x + 8$

$m = -\dfrac{3}{1} \qquad (0,8)$

(iii) $2x + y = 4$

$y = -2x + 4$

$m = -\dfrac{2}{1} \qquad (0,4)$

(iv) $3x + y = 5$

$y = -3x + 5$

$m = -\dfrac{3}{1} \qquad (0,5)$

(v) $2x - y = 3$

$-y = -2x + 3$

$y = 2x - 3$

$(0,-3) \qquad m = \dfrac{2}{1}$

(vi) $2x - y = 4$

$-y = -2x + 4$

$y = 2x - 4$

$(0,-4) \qquad m = \dfrac{2}{1}$

(vii) $5x + y = 7$

$y = -5x + 7$

$m = -\dfrac{5}{1} \qquad (0,7)$

When $y$ has a coefficient, divide every term across.

(viii) $3x - 2y = 8$

$-2y = -3x + 8$

$\dfrac{2y}{2} = \dfrac{3x}{2} - \dfrac{8}{2}$

$m = \dfrac{3}{2} \qquad (0,-4)$

(ix) $3x - 4y = 6$

$-4y = -3x + 6$

$\dfrac{4y}{4} = \dfrac{3x}{4} - \dfrac{6}{4}$

$y = \dfrac{3}{4}x - \dfrac{6}{4}$

$m = \dfrac{3}{4}$

(x) $7x - 2y = 9$

$-2y = -7x + 9$

$2y = 7x - 9$

$y = \dfrac{7}{2}x - \dfrac{9}{2}$

$m = \dfrac{7}{2}$

Section 2 of 4

Drawing a line

On a Cartesian plane draw (i) $x = 1$ (ii) $y = 2$

Remember

1.$y = $ horizontal.

2.$x = $ vertical.

Draw $2x + y = 4$ (use $y = mx + c$)

$y = -2x + 4$

$m = -\dfrac{2}{1} \qquad (0,4)$

Draw $3x - y = 1$

$-y = -3x + 1$

$y = 3x - 1$

$m = \dfrac{3}{1} \qquad (0,-1)$

Draw $3x + 4y = 12$

$\dfrac{4y}{4} = \dfrac{-3x}{4} + \dfrac{12}{4}$

$y = -\dfrac{3}{4}x + 3$

$(0,3) \qquad m = -\dfrac{3}{4}$

Section 3 of 4

Drawing with intercepts

Find where the line cuts each axis: let $x = 0$, then let $y = 0$.

(i) $x + 2y = 4$

$x = 0: \quad 2y = 4, \quad y = 2 \quad \to (0,2)$

$y = 0: \quad x = 4 \quad \to (4,0)$

(ii) $x + y = 3$

$x = 0: \quad y = 3 \quad \to (0,3)$

$y = 0: \quad x = 3 \quad \to (3,0)$

(iii) Draw $2x + 3y = 6$

$x$-axis $(y = 0): \quad 2x = 6, \quad x = 3 \quad \to (3,0)$

$y$-axis $(x = 0): \quad 3y = 6, \quad y = 2 \quad \to (0,2)$

Section 4 of 4

Parallel lines

Method

1.Parallel $\Rightarrow$ same slope.

2.Find the slope from $y = mx + c$, then use the given point.

(i) Through $(1,3)$ parallel to $2x + y = 7$

$y = -2x + 7 \quad \to m = -2$

$(1,3) \to x_1,y_1$

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

(ii) Through $(1,5)$ parallel to $5x + y = 7$

$y = -5x + 7 \quad \to m = -5$

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

(iii) Through $(2,-1)$ parallel to $3x - y = 6$

$-y = -3x + 6, \quad y = 3x - 6 \quad \to m = 3$

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

(iv) Through $(3,-5)$ parallel to $7x - y = 6$

$-y = -7x + 6, \quad y = 7x - 6 \quad \to m = 7$

$y + 5 = 7(x - 3)$

(v) Through $(1,2)$ parallel to $6x - y = 8$

$-y = -6x + 8, \quad y = 6x - 8 \quad \to m = 6$

$y - 2 = 6(x - 1)$

(vi) Through $(1,-2)$ parallel to $3x + y = 7$

$y = -3x + 7 \quad \to m = -3$

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

$y + 2 = -3x + 3$

$3x + y = 1$

(vii) Through $(1,-3)$ parallel to $5x + y = 6$

$y = -5x + 6 \quad \to m = -5$

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

$y + 3 = -5x + 5$

$5x + y = 2$

SUM

The lot in one box

Slope, drawing & parallel toolkit

1.Slope: write $y = mx + c$; read $m$ and $(0,c)$. Coefficient on $y$? Divide across.

2.$y =$ horizontal, $x =$ vertical.

3.Draw by $y = mx + c$, or by intercepts (let $x=0$, then $y=0$).

4.Parallel $\Rightarrow$ same slope; then use the point.

End of lesson

Slope, Drawing & Parallel Lines — HL · Mathslive.ie