91Ó°ÊÓ

When graphing linear equations, the **x-intercept** tells us where the line crosses the x-axis. This point occurs when the value of y is zero. To find the x-intercept of an equation, we set the y variable to zero and solve for x.

For example, if we take the equation from our exercise, which has been converted into slope-intercept form as \(y = \frac{1}{2}x\), we find the x-intercept by substituting \(y = 0\). This becomes \(0 = \frac{1}{2} x\). Solving for x, we multiply both sides by 2, giving us \(x = 0\).

Hence, the x-intercept is at the point \((0,0)\), which means the line crosses the x-axis directly at the origin.

The **y-intercept** is where the line crosses the y-axis. At this point, the value of x is zero. To locate the y-intercept, we set x to zero and solve the equation for y.

In our exercise, using the equation \(y = \frac{1}{2}x\), we substitute \(x = 0\) to find the y-intercept. This becomes \(y = \frac{1}{2} \times 0\). The result is \(y = 0\).

Consequently, the y-intercept is also at the origin \((0, 0)\). This implies the line intersects the y-axis directly at the origin, which is a special case where both intercepts coincide.

The **slope-intercept form** of a linear equation is a way of writing an equation so that the slope and intercept are immediately available. The standard format is \(y = mx + b\). Here, \(m\) denotes the slope, and \(b\) represents the y-intercept.

In the case of our exercise, we converted the equation \(x = 2y\) into slope-intercept form, yielding \(y = \frac{1}{2}x + 0\). This reveals that the slope \(m\) is \(\frac{1}{2}\) and the y-intercept \(b\) is 0.

The slope tells us how steep the line is, indicating that for every 2 units moved horizontally, the line rises 1 unit vertically. By incorporating both the y-intercept and the slope, we can easily sketch the line on a graph, even when it starts or ends at the origin \((0,0)\).