Problem 55 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the line. Graph the equation. Label the points where the line crosses the axes. $$ 0.5 y=-2 x+8 $$

Step-by-Step Solution

Answer

The x-intercept is 4 and the y-intercept is 16. These points are plotted on the graph and a line passing through them is drawn.

Step 1: Finding the x-intercept

To find the x-intercept, set \(y = 0\) in the given equation, \(0.5 y = -2x +8\). This gives \(0 = -2x +8\). Solving this for \(x\) will give us the x-intercept.

Step 2: Finding the y-intercept

To find the y-intercept, set \(x = 0\) in the given equation, \(0.5 y = -2x + 8\). This gives \(0.5 y = 8\). Solving this for \(y\) will give us the y-intercept.

Step 3: Graphing and labelling

Now, plot these intercepts in a graph and draw a line passing through the intercepts. Finally, label the x-intercept and y-intercept on the graph.

Key Concepts

Understanding the X-Intercept

The x-intercept of a line is the point where the line crosses the x-axis. At this specific point, the value of y is always zero. When you're given a linear equation like \(0.5y = -2x + 8\), finding the x-intercept involves substituting \(y = 0\) into the equation.
By doing so, the equation simplifies to \(0 = -2x + 8\). Solving for \(x\) involves isolating \(x\) on one side, typically through addition or subtraction, followed by division if necessary. In this exercise, once you isolate \(x\), you'll find that \(x = 4\). So, the x-intercept is at the point \((4, 0)\).
Identifying the x-intercept is crucial because it's one of the two main "anchor" points for graphing the line efficiently, alongside the y-intercept.

Delving into the Y-Intercept

The y-intercept is the point where the line intersects the y-axis. At this point, the value of x is always zero. For the equation \(0.5y = -2x + 8\), to find the y-intercept, set \(x = 0\).
Substituting \(x = 0\) into the equation, you have \(0.5y = 8\).
Solving for \(y\) by dividing both sides by 0.5 gives \(y = 16\).
Therefore, the y-intercept is at the point \((0, 16)\).
This point is vital as it provides the starting point on the y-axis when graphing the line, along with the x-intercept point to draw the line accurately.

Graphing Lines with Intercepts

Graphing lines using intercepts is a straightforward technique that allows you to easily sketch the line on a graph. Start by plotting both the x-intercept and the y-intercept on the Cartesian plane.

• The x-intercept from our equation is \((4, 0)\)
• The y-intercept is \((0, 16)\)

With these intercepts, draw a straight line passing through both points. Make sure to use a ruler for precision. This visual representation helps in understanding the behavior and direction of the line.
Label the intercepts on the graph to clearly indicate where the line intersects each axis. This step not only aids in comprehension but also is a critical part of correctly completing graphing exercises in mathematics.