Problem 67 - College Algebra with Modeling and Visualization

Question

Determine the \(x\) - and \(y\) -intercepts on the graph of the equation. Graph the equation. $$ y=8 x-5 $$

Step-by-Step Solution

Answer

The x-intercept is \( \left( \frac{5}{8}, 0 \right) \) and the y-intercept is \( (0, -5) \).

Step 1: Determine the x-intercept

To find the x-intercept, set \( y = 0 \). Substitute \( y = 0 \) into the equation \( y = 8x - 5 \), resulting in \( 0 = 8x - 5 \). Solve for \( x \) by adding 5 to both sides and then dividing by 8. The equation becomes \( 8x = 5 \), and \( x = \frac{5}{8} \). So, the x-intercept is \( \left( \frac{5}{8}, 0 \right) \).

Step 2: Determine the y-intercept

To find the y-intercept, set \( x = 0 \). Substitute \( x = 0 \) into the equation \( y = 8x - 5 \), resulting in \( y = 8(0) - 5 \), which simplifies to \( y = -5 \). Therefore, the y-intercept is \( (0, -5) \).

Step 3: Graph the Equation

Now that we have the intercepts, plot the points \( \left( \frac{5}{8}, 0 \right) \) and \( (0, -5) \) on the graph. Connect these two points with a straight line. This line represents the graph of the equation \( y = 8x - 5 \).

Key Concepts

Understanding the x-intercept

The x-intercept of a linear equation is where the graph crosses the x-axis. To find this intercept, we set the value of y to zero and solve the equation to find the corresponding x-value. Consider the equation provided: \( y = 8x - 5 \).

• First, substitute \(y = 0\) into the equation, giving us \(0 = 8x - 5\).
• Next, solve for x by adding 5 to both sides: \(8x = 5\).
• Finally, divide both sides by 8 to isolate x: \(x = \frac{5}{8}\).

Therefore, the x-intercept is at the point \( \left( \frac{5}{8}, 0 \right) \). This indicates that if you follow the line to where it crosses the x-axis, it will be almost at the \(\frac{5}{8}\) mark. The intercept tells us a lot about the start of our graph on the horizontal axis.

Understanding the y-intercept

The y-intercept occurs where the graph crosses the y-axis. To discover this point, set the value of x to zero in the given equation and solve for y. With the equation \( y = 8x - 5 \), let's find it:

• Substitute \(x = 0\) into the equation: \( y = 8(0) - 5 \).
• Simplify to find the value of y: \( y = -5 \).

Thus, the y-intercept is at \( (0, -5) \). This point tells us that when the graph crosses the y-axis, it does so at -5. This is your starting point for the line on the vertical axis and signifies where the line will touch the y-axis when graphing.

Graphing lines with intercepts

Graphing a line using the intercepts involves plotting the points found from the x-intercept and the y-intercept onto a graph. Once these points are plotted, they can be connected to form a straight line. Let's walk through this:

• First, mark the x-intercept \( \left( \frac{5}{8}, 0 \right) \) on your graph. Remember, it's slightly more than halfway to 1 on the x-axis.
• Then, mark the y-intercept \( (0, -5) \). Place this on the y-axis, down five units.
• Draw a straight line through these two points, which represents the equation \( y = 8x - 5 \).

The line should pass smoothly through both intercepts, providing a visual understanding of the equation. Graphing in this way with intercepts is straightforward and visually appealing, as it makes understanding the slope and direction of the line easy.