Problem 55 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(x-\frac{1}{2} y=8\)

Step-by-Step Solution

Answer

The x-intercept is (8, 0), and the y-intercept is (0, -16).

Step 1: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). Substitute \(y = 0\) into the equation: \(x - \frac{1}{2}(0) = 8\). Simplifying gives \(x = 8\). Thus, the x-intercept is the point \((8, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). Substitute \(x = 0\) into the equation: \(0 - \frac{1}{2}y = 8\). Simplifying gives \(-\frac{1}{2}y = 8\). Multiply both sides by \(-2\) to get \(y = -16\). Thus, the y-intercept is the point \((0, -16)\).

Step 3: Graph the Equation

Plot the x-intercept \((8, 0)\) and the y-intercept \((0, -16)\) on the coordinate plane. Draw a straight line through these two points to represent the equation \(x - \frac{1}{2}y = 8\). This line is the graph of the equation.

Key Concepts

x-intercept

In linear equations, the x-intercept is the point where the graph of the equation crosses the x-axis. This means that it is the value of \(x\) when \(y\) is zero. To find the x-intercept in any equation, we substitute \(y = 0\) and solve for \(x\).
For example, with the equation \(x - \frac{1}{2}y = 8\), when \(y=0\), the equation becomes \(x - \frac{1}{2}(0) = 8\). Simplifying this gives \(x = 8\). Hence, the x-intercept here is \((8, 0)\).
Knowing how to find the x-intercept helps visualize where a line starts or stops on the horizontal axis. It's a fundamental step in graphing linear equations as it represents one of the anchor points on the graph.

y-intercept

The y-intercept is where the graph of an equation intersects the y-axis. At this interception, the value of \(x\) is zero. The y-intercept helps describe the vertical position of the line on the graph, providing a reference point against which the slope is perceived.
To identify the y-intercept, make \(x = 0\) in the equation and solve for \(y\). In our exercise, substituting \(x = 0\) in \(x - \frac{1}{2}y = 8\) results in \(0 - \frac{1}{2}y = 8\). Simplifying gives \(-\frac{1}{2}y = 8\), and multiplying both sides by \(-2\) yields \(y = -16\). Thus, the y-intercept is the point \((0, -16)\).
Using the y-intercept is essential in graphing because it provides an exact point through which the line passes, assisting in ensuring the line's accuracy and slope.

graphing equations

Graphing equations involves translating algebraic expressions into visual forms by drawing on a coordinate plane. The goal is to visualize the relationship between variables as a line or curve, depending on the equation type, making it easier to understand patterns or predict behaviors.
For a linear equation like \(x - \frac{1}{2}y = 8\), start by plotting the x-intercept \((8, 0)\) and the y-intercept \((0, -16)\). These are the points where the equation interacts with the axis, and by joining them with a straight line, you represent all the solutions of the equation.

• Plot the x-intercept first, by marking the point on the x-axis.
• Next, plot the y-intercept by marking the point on the y-axis.
• Use a ruler to ensure a straight line through these points to show the line's path.

Graphing equations allows for an immediate understanding of how changes in one variable might influence the other, providing insights into solutions visually.