Equations of Straight Lines (VCE SSCE General Mathematics): Revision Notes

Worked Example: Finding Slope Using the Formula

Question: Find the slope of the line passing through $(1, 7)$ and $(4, 2)$ using the slope formula. Give your answer to two decimal places.

Solution:

Using the formula: $\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$

Let $(x_1, y_1) = (1, 7)$ and $(x_2, y_2) = (4, 2)$

Substituting these values: $\text{slope} = \frac{2 - 7}{4 - 1}$

$= \frac{-5}{3}$

$= -1.67 \text{ (to 2 d.p.)}$

The negative slope indicates that the line is decreasing (going downward from left to right).

Worked Example: Identifying Intercept and Slope

Question: Write down the y-intercept and slope of each straight line:

• a) $y = -6 + 9x$
• b) $y = 10 - 5x$
• c) $y = -2x$
• d) $y - 4x = 5$

Solution:

For each equation, we need to ensure it's in intercept-slope form $y = a + bx$, then identify the values.

a) $y = -6 + 9x$

This is already in intercept-slope form.

• y-intercept = $-6$
• slope = $9$

b) $y = 10 - 5x$

This can be written as $y = 10 + (-5)x$

• y-intercept = $10$
• slope = $-5$

c) $y = -2x$

This can be written as $y = 0 + (-2)x$ or $y = 0 - 2x$

When there is no constant term, the y-intercept is zero.

• y-intercept = $0$
• slope = $-2$

d) $y - 4x = 5$

First, rearrange into intercept-slope form by making $y$ the subject: $y = 5 + 4x$

Now we can identify:

• y-intercept = $5$
• slope = $4$

Worked Example: Constructing Equations

Question: Write down the equations of straight lines with the following y-intercepts and slopes:

• a) y-intercept = $9$, slope = $6$
• b) y-intercept = $2$, slope = $-5$
• c) y-intercept = $-3$, slope = $2$

Solution:

Using the form $y = a + bx$, substitute the given values for $a$ (y-intercept) and $b$ (slope).

a) y-intercept = $9$, slope = $6$

Substituting $a = 9$ and $b = 6$: $y = 9 + 6x$

b) y-intercept = $2$, slope = $-5$

Substituting $a = 2$ and $b = -5$: $y = 2 + (-5)x$

This can also be written as: $y = 2 - 5x$

c) y-intercept = $-3$, slope = $2$

Substituting $a = -3$ and $b = 2$: $y = -3 + 2x$

This can also be written as: $y = 2x - 3$

Worked Example: Sketching a Graph from its Equation

Question: Sketch the graph of $y = 8 + 2x$

Solution:

Step 1: Identify the equation $y = 8 + 2x$

Step 2: Find the y-intercept

The equation is in intercept-slope form, so the y-intercept is the constant term: $\text{y-intercept} = 8$

This gives us the point $(0, 8)$

Step 3: Find a second point

Choose a convenient value for $x$. Let's use $x = 5$:

When $x = 5$: $y = 8 + 2(5)$ $y = 8 + 10$ $y = 18$

So $(5, 18)$ is a point on the line.

Step 4: Draw the graph

• Draw and label the axes
• Mark the two points: $(0, 8)$ and $(5, 18)$
• Draw a straight line through these points
• Label the line with its equation