Parallel Lines (Grade 11 NSC Matric Mathematics): Revision Notes

Practical Example: Parallel Lines with Different Y-Intercepts

The diagram shows two parallel lines: $y = 2x + 3$ and $y = 2x - 1$.

Notice both have:

• Gradient = 2 (same slope)
• Different y-intercepts (3 and -1)

This is what makes them parallel - same direction, different positions.

Worked Example: Finding Parallel Line Through Given Point

Question: Determine the equation of the line that passes through the point $(-1; 1)$ and is parallel to the line $y - 2x + 1 = 0$.

Solution:

Step 1: Write the equation in gradient-intercept form

• $y - 2x + 1 = 0$
• $y = 2x - 1$

The gradient is m = 2.

Step 2: Apply the parallel line rule

Since the lines are parallel: m₁ = m₂ = 2

Step 3: Use gradient-point form

• $y - y_1 = m(x - x_1)$

Substitute $m = 2$ and point $(-1; 1)$:

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

Answer: The equation is y = 2x + 3

Worked Example: Parallel Line Using Angle of Inclination

Question: Line $AB$ passes through point $A(0; 3)$ and has an angle of inclination of $153.4°$. Determine the equation of line $CD$ which passes through point $C(2; -3)$ and is parallel to $AB$.

Solution:

Step 1: Find the gradient using angle of inclination

• $m_{AB} = \tan θ$
• $m_{AB} = \tan 153.4°$
• $m_{AB} = -0.5$

Step 2: Apply parallel lines rule

Since $AB \parallel CD$:

• $m_{CD} = m_{AB} = -0.5$

Step 3: Use gradient-point form $y - y_1 = m(x - x_1)$

Substitute $m = -\frac{1}{2}$ and point $(2; -3)$:

Answer: The equation is y = -½x - 2