Problem 34 - Introductory and Intermediate Algebra for College Students 4th

Question

Use intercepts and a checkpoint to graph each equation. $$3 x-6 y=15$$

Step-by-Step Solution

Answer

The x-intercept is (5,0), the y-intercept is (0,-2.5). After verifying, the checkpoint (1,1) is on the line. Hence, the graph of the equation \(3x - 6y = 15\) is a straight line passing through these points.

Step 1: Find the x-intercept

Set \(y=0\) and solve for \(x\) in the equation \(3x - 6y = 15\). This gives the x-intercept.

Step 2: Find the y-intercept

Set \(x=0\) and solve for \(y\) in the equation \(3x - 6y = 15\). This gives the y-intercept.

Step 3: Choose a checkpoint

Select a point not on the line obtained by using the intercepts. This point is referred to as the checkpoint. In this scenario, we can take (1,1) for ease.

Step 4: Verify the checkpoint

Substitute the coordinates of the checkpoint back into the original equation. These should satisfy the equation, if not, recheck the calculations done previously.

Step 5: Plot the intercepts and the checkpoint on the graph

Now plot these points on the graph (the x-intercept, the y-intercept, and the checkpoint). A straight line has now been defined by these points.

Step 6: Draw the line

Draw a line through the three points to complete the graph of the equation.

Key Concepts

X-Intercept

The x-intercept of a line is the point at which the line crosses the x-axis. This is where the graph of the equation has a y-value of zero. To find the x-intercept algebraically, you set the y-variable to zero and solve for x.

For example, given the equation 3x - 6y = 15, to find the x-intercept, replace y with zero: 3x - 6(0) = 15, which simplifies to 3x = 15. Dividing both sides by 3, you find that x = 5. Therefore, the x-intercept is the point (5,0). This specific intercept can tell us that when a graph passes through this point, the value of y is null at the location where x equals 5.

Y-Intercept

Conversely, the y-intercept is found where a line crosses the y-axis; here the line has an x-value of zero. To locate the y-intercept on an equation, you set the x-variable to zero and solve for y.

Using the same equation, set x=0 in 3x - 6y = 15. The equation simplifies to -6y = 15. Dividing by -6 yields y = -2.5. The coordinates of the y-intercept are therefore (0,-2.5). A line will intersect the y-axis at this point, indicating that when x is zero, y will have a value of -2.5.

Checkpoint Method

The checkpoint method involves choosing a random point (not on the x or y-axes) and verifying that it satisfies the equation. This helps to confirm that you've correctly found the intercepts and can be very helpful in ensuring the accuracy of your graph.

For instance, using the checkpoint (1,1) for the equation 3x - 6y = 15 and substituting these values in gives us 3(1) - 6(1) = 3 - 6, which equals -3 and not 15, meaning the checkpoint does not lie on the line. This indicates either the checkpoint is incorrect, or the previous steps have errors. It is essential to select a suitable checkpoint and make sure all the points align perfectly on the graph to depict the line accurately.