Problem 86 - Introductory and Intermediate Algebra for College Students 4th

Question

Use intercepts to graph \(3 x-5 y=15\) (Section \(3.2,\) Example 4 )

Step-by-Step Solution

Answer

x-intercept: 5, y-intercept: -3. The graph is a straight line passing through points (5,0) and (0,-3).

Step 1: Find the x-intercept

To find the x-intercept set \(y = 0\) in the equation \(3x - 5y = 15\). This gives \(3x = 15\), which simplifies to \(x = 5\). So, the x-intercept is 5.

Step 2: Find the y-intercept

To find the y-intercept set \(x = 0\) in the equation \(3x - 5y = 15\). This gives \(-5y = 15\), which simplifies to \(y = -3\). So, the y-intercept is -3.

Step 3: Graphing the line

On a graph, plot the points (5,0) and (0,-3) which are the x-intercept and y-intercept respectively. Draw a straight line passing through these points. This line representa the linear equation \(3x - 5y = 15\).

Key Concepts

X-Intercept

Understanding the concept of the x-intercept is essential when graphing linear equations. An x-intercept is a point on the graph where the line crosses the x-axis. At this point, the value of y is always zero since it's the horizontal axis of the graph. To find the x-intercept, you set the y variable in the equation to zero and solve for x. For example, in the equation \(3x - 5y = 15\), setting \(y = 0\) results in \(3x = 15\), and dividing both sides by 3 gives us \(x = 5\). Therefore, the x-intercept is the point (5,0).

When graphing, you would mark this point on the horizontal axis. Knowing the x-intercept offers a starting point to draw the line representing the linear equation. This single point can guide you in checking the slope and direction of the line as you plot additional points or draw the graph.

Y-Intercept

The y-intercept is another critical point used in graphing linear equations, and it refers to the place where the line crosses the y-axis. The y-intercept has an x-value of zero because it’s located on the vertical axis. To determine the y-intercept of a linear equation, you set the x variable to zero and then solve for y. Taking the same linear equation, \(3x - 5y = 15\), and substituting \(x = 0\), we get \(-5y = 15\). Solving for y by dividing by -5, we find \(y = -3\). So, the y-intercept is the point (0,-3).

This intercept is particularly helpful since it's where you start plotting the graph on the vertical axis. Plotting the y-intercept allows you to visualize where the line will be oriented in relation to the y-axis. This is especially useful if you need to sketch the graph by hand or when using the slope to find other points on the grid.

Plotting Points on a Graph

Plotting points on a graph is crucial to visualizing the relationship between variables in a linear equation. Once you have the intercepts, you can begin plotting. Start with the x-intercept and y-intercept. In our example, these would be the points (5,0) and (0,-3). Plot them precisely where they belong on the x and y axes respectively.

After marking the intercepts, it's helpful to plot additional points to ensure the line is straight. You can do this by choosing arbitrary values for x or y and solving the equation for the other variable. Then, connect the dots with a straight edge, ensuring the line passes through both intercepts.

Consistent Plotting

Always use a ruler or another straight edge to draw the line through the points. This ensures accuracy and a clear visual representation of the linear equation. A well-drawn graph can be a valuable tool for analyzing the equation further, predicting values, and understanding the slope and rate of change between variables.