Problem 36 - Introductory and Intermediate Algebra for College Students 4th

Question

Use intercepts and a checkpoint to graph each equation. $$2 x+y=0$$

Step-by-Step Solution

Answer

The equation \(2x + y = 0\) is graphed by identifying the intercepts at (0,0) and verifying the point (1, -2). The line passing through these points represents the equation.

Step 1: Finding the Y-intercept

To find the y-intercept, set \(x = 0\) in the equation. That provides the y-coordinate of the y-intercept. In this case, \(2(0) + y = 0\) simplifies to \(y = 0\). Thus, the y-intercept is at the point (0,0).

Step 2: Finding the X-intercept

To find the x-intercept, set \(y = 0\) in the equation. That provides the x-coordinate of the x-intercept. In this case, \(2x + 0 = 0\) simplifies to \(2x = 0\), and solving for x gives \(x = 0\). Thus, the x-intercept is also at the point (0,0).

Step 3: Select and Verify a Checkpoint

Choose a checkpoint that is not on the axes. For instance, select the point (1, -2) to verify if it satisfies the equation.Since substituting \(x = 1\) and \(y = -2\) in the equation gives \(2(1) + (-2) = 0\), this means that it is true. Therefore, the point (1, -2) lies on the line.

Step 4: Graph the Equation

Now plot the points (1, -2) and (0,0) on the graph. Draw a straight line passing through these points. This line represents the equation \(2x + y = 0\).

Key Concepts

X-intercept

The x-intercept of a line on a graph is the point where the line crosses the x-axis. To find this point, you set the y-value to zero in the linear equation, because on the x-axis, y is always zero. For example, let's consider the equation given in the exercise: \[2x + y = 0\]By setting \(y = 0\), the equation becomes:\[2x + 0 = 0\]This simplifies to \(2x = 0\), which can be solved to give \(x = 0\). Therefore, in this example, the x-intercept is the point (0, 0).

• If the x-intercept is (0, 0), the graph crosses the x-axis at the origin.
• To verify, always substitute the x-intercept back into the original equation.

Y-intercept

The y-intercept is the point where the line meets the y-axis, and at this point, the x-coordinate is always zero. By substituting \(x = 0\) into the equation, you solve for the y-value. Let's see this with the given equation:\[2x + y = 0\]Putting \(x = 0\) gives:\[2(0) + y = 0\]Which simplifies to \(y = 0\). Thus, the y-intercept here is also the origin, at the point (0, 0).

• Remember, the y-intercept indicates where the line crosses the y-axis.
• If it's at the origin, the line passes through both axes there.

Slope-intercept form

The slope-intercept form is a useful way of writing a linear equation because it directly shows the slope and y-intercept of a line. It is usually expressed as:\[y = mx + b\]Here, \(m\) is the slope and \(b\) is the y-intercept. From the given equation:\[2x + y = 0\]We can rearrange it to match the slope-intercept form by solving for \(y\):\[y = -2x + 0\]This expression makes it clear that the slope \(m\) is \(-2\) and the y-intercept \(b\) is 0.

• This form lets you quickly identify how steep a line is and where it starts on the y-axis.
• Knowing the slope helps predict how the line behaves as you move along the x-axis.

Checkpoint method

The checkpoint method is an effective way to confirm that you've graphed a line correctly. After placing the intercepts on the graph, select another point, called a "checkpoint," which is not an intercept, and see if it satisfies the equation.In this exercise, after plotting the intercepts, the point (1, -2) is chosen as a checkpoint:\[2(1) + (-2) = 0\]This calculation is true, showing that (1, -2) lies on the line.

• Choose a checkpoint that is easy to work with.
• Verify it satisfies the equation to ensure the line is accurately drawn.

By ensuring your checkpoint checks out, you gain confidence that the entire graph is accurate.