Understanding the equations of straight-line graphs is foundational in coordinate geometry. This section delves into interpreting and formulating linear equations, emphasizing various forms such as $y = mx + c$ and $ax + by = c$, focusing on identifying the gradient (slope) and y-intercept.

Image courtesy of HMH

What is a Linear Equation?

A linear equation represents a straight line on a coordinate plane, typically expressed in the form $y = mx + c$, where:

• $m$ is the gradient, determining the steepness and direction of the line.
• $c$ is the y-intercept, indicating where the line crosses the y-axis.

Another common form is $ax + by = c$, which can be rearranged into $y = mx + c$ or other required forms.

Interpreting Straight-Line Graphs

To interpret straight-line graphs, one must grasp two main components: the gradient and the y-intercept.

• Gradient (m): Shows the change in $y$ for a unit change in $x$. Positive gradients indicate an upward slope, negative gradients a downward slope, and a zero gradient indicates a horizontal line.
• Y-intercept (c): The point where the line intersects the y-axis, determined by the $y$ value when $x = 0$.

Formulating Linear Equations

Creating equations of straight lines involves finding the gradient and y-intercept from a graph or set of points.

From a Graph

Directly identify the y-intercept where the line crosses the y-axis. To find the gradient, select two points on the line, applying:

From Points

Given two points $(x_1, y_1)$ and $(x_2, y_2)$, calculate the gradient using the same formula for $m$. Once $m$ is known, use $y = mx + c$ with one point to solve for $c$.

Image courtesy of CueMath

Worked Examples

Example 1: Finding the Equation from a Gradient and a Point

Given a gradient $m = -3$ and a point $(-2, 3)$, find the equation of the line.

Solution:

1. Using the formula $y = mx + c$, we substitute $m = -3$ and the point $(-2, 3)$ to solve for $c$:

2. Solving for $c$, we find $c = -3$.

Thus, the equation of the line is:

Example 2: Determining Gradient and Y-intercept from an Equation

Given the equation $3x - 4y = 12$, rearrange it to the form $y = mx + c$ to identify the gradient and y-intercept.

Solution:

1. Rearrange the equation:

2. Gradient ($m$): $-\dfrac{3}{4}$

3. Y-intercept ($c$): $3$