Problem 34 - Introductory Algebra for College Students

Question

Use intercepts and a checkpoint to graph equation. \(3 x-6 y=15\)

Step-by-Step Solution

Answer

The line for the equation \(3x - 6y = 15\) can be graphed using the X-intercept (5,0), the Y-intercept (0, -2.5) and the checkpoint (3,0).

Step 1: Find the X-intercept

Set y = 0 in the equation and solve for x. This can be done by substituting y with 0 in the given equation. So, the equation becomes 3x - 6(0) = 15. Solving this, we get x = 5.

Step 2: Find the Y-intercept

Set x = 0 in the equation and solve for y. This can be done by substituting x with 0 in the given equation. So, the equation becomes 3(0) - 6y = 15. Solving this, we get y = -2.5.

Step 3: Choose a checkpoint

Choose a point (x, y) that is not the X or Y intercept. A convenient point to pick is often (1,1) unless that is one of the intercepts. Substitute the values of the chosen point in the equation. The equation then becomes 3(1) - 6(1) = 15. The result is -3 which is not equal to 15, so (1,1) is not a point on the line. So, if (1,1) doesn't fit, choose another convenient point like (2,1). The equation then becomes 3(2) - 6(1) = 15. The result again is not equal to 15. Repeat this process until a valid point is found. The point (3,0) fulfills the equation, so, (3,0) is a valid checkpoint on the line.

Step 4: Graph the line

Using the X-intercept (5,0), Y-intercept (0, -2.5), and the checkpoint (3,0), graph the line. Those points can be plotted on the graph and then connected with a straight line.

Key Concepts

Intercepts in Graphing

Intercepts are crucial tools in graphing linear equations because these points reveal where the graph intersects the axes. When you have a linear equation, such as \(3x - 6y = 15\), finding the intercepts simplifies the process of sketching the line on a coordinate plane.

• The intercepts provide clear anchor points, making it easier to draw the graph accurately.
• They are typically used with other points, such as checkpoints, to help ensure the entire line is plotted correctly.

Using intercepts effectively begins with knowing how to calculate both the x-intercept and y-intercept, setting up the basis for fully understanding linear graphs.

X-Intercept

The x-intercept is the point where the graph crosses the x-axis. This is easily found by setting the y value to zero, because at the x-axis point, y is always zero.
For the equation \(3x - 6y = 15\), when y is set to zero, the equation becomes\(3x = 15\). Solving for x gives us \(x = 5\), meaning the x-intercept is the point \((5,0)\).

• Remember, to find the x-intercept, substitute \(y = 0\) and solve for \(x\).
• The x-intercept is always easy to plot at \((x, 0)\).

This intercept provides one of the basic anchor points for drawing and verifying the line accurately on the graph.

Y-Intercept

The y-intercept is where the graph crosses the y-axis. This point is essential because it's one of the easiest spots to find and plot for a linear equation.
To determine the y-intercept, set the x value to zero in the equation \(3x - 6y = 15\), thus the equation becomes \(-6y = 15\). Solving for y yields \(y = -2.5\), making the y-intercept the point \((0, -2.5)\).

• Finding the y-intercept involves replacing \(x = 0\) and solving for \(y\).
• The y-intercept is just \((0, y)\), another simple-to-plot reference point.

The y-intercept is a vital part of graphing as it showcases where the line begins in relation to the y-axis.

Checkpoints in Graphs

Checkpoints are additional points used to confirm that a line drawn using intercepts is accurately plotted. While intercepts provide great starting points, a checkpoint gives extra assurance of correctness.
Selecting a good checkpoint involves choosing a coordinate not lying on either axis, such as \((3, 0)\). Substituting these values into the equation \(3x - 6y = 15\) should validate the line equation if plotted correctly.

• Checking involves substituting the \(x ext{- and }y\) values into the equation.
• If the equation holds true with the checkpoint values, it confirms the graph's accuracy.
• Choose points that offer simple calculations.

Using checkpoints ensures that beyond just connecting the intercepts, the whole line satisfactorily represents the equation and its linear relationship.