In this section, we will explore how to find the gradient and equation of a line parallel to a given line, with a practical example to aid your understanding. Understanding parallel lines is crucial in coordinate geometry, as it allows us to analyse and solve a variety of geometric problems.

Understanding the Gradient of Parallel Lines

The gradient (or slope) of a line measures how steep the line is. It's a crucial concept in coordinate geometry, as it helps us understand the direction and steepness of lines.

• Key Concept: Parallel lines have equal gradients. This means that if two lines are parallel, their gradients will be the same.
• To find the gradient of a line, we use the formula:

where $m$ is the gradient, and $(x_1, y_1)$ and $(x_2, y_2)$ are any two points on the line.

Image courtesy of CueMath

Finding the Equation of a Line

To find the equation of a line, you need to know the gradient and a point through which the line passes. The general form of the equation of a line is $y = mx + c$, where $m$ is the gradient and $c$ is the y-intercept.

• Point-Slope Form: Another useful form, especially when given a point through which the line passes and its gradient, is the point-slope form: $y - y_1 = m(x - x_1)$, where $m$ is the gradient and $(x_1, y_1)$ is the point.

Example: Finding a Parallel Line

Let's apply these concepts to find a line parallel to $y = 4x - 1$ that passes through the point $(1, -3)$.

Step 1: Identify the Gradient of the Given Line

The given line is $y = 4x - 1$. The coefficient of $x$ is $4$, which means the gradient of the line is $4$.

Step 2: Use the Gradient for the Parallel Line

Since parallel lines have the same gradient, the parallel line we want to find will also have a gradient of $4$.

Step 3: Use the Point-Slope Form

We have a point $(1, -3)$ and a gradient $4$. Plugging these into the point-slope form gives us:

Step 4: Simplify to Get the Equation

To get the equation in the form $y = mx + c$, we simplify the above equation:

Therefore, the equation of the line parallel to $y = 4x - 1$ and passing through the point $(1, -3)$ is $y = 4x - 7$.

Practice Questions

Question 1:

Find the equation of a line parallel to $y = 2x + 5$ that passes through $(-2, 3)$.

Solution:

• Given line: $y = 2x + 5$
• Gradient: $2$
• Parallel line's gradient: $2$
• Point given: $(-2, 3)$
• Using the point-slope form: $y - 3 = 2(x + 2)$

Simplifying:

Question 2:

Determine the equation of a line parallel to $y = -3x + 4$ and passing through the point $(4, -2)$.

Solution:

• Given line: $y = -3x + 4$
• Gradient: $-3$
• Parallel line's gradient: $-3$
• Point given: $(4, -2)$
• Using the point-slope form: $y + 2 = -3(x - 4)$

Simplifying:

Key Takeaways

• Parallel lines have the same gradient.
• The equation of a line can be found using the gradient and a point it passes through.
• Use the point-slope form $y - y_1 = m(x - x_1)$ to easily find the equation of a line when given a point and gradient.