Problem 35 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The x-intercept is (0,0), the y-intercept is also (0,0), and a checkpoint is at (1,-0.5). By plotting these points and drawing a straight line through, we get the graph of the equation \(x + 2y = 0\).

Step 1: Find the x-intercept

We set \(y = 0\) in the equation to find the x-intercept. This gives us the equation \(x + 2(0) = x = 0\). So, the x-intercept is at the point (0,0).

Step 2: Find the y-intercept

We set \(x = 0\) in the equation to find the y-intercept. This gives us \(0 + 2y = y\), from which we get \(y = 0\). So, the y-intercept is also at the point (0,0).

Step 3: Find a checkpoint

The checkpoint is a third point on the graph that we can use to verify our graph. Set \(x = 1\) and solve for \(y\) to find this point. Substituting \(x = 1\) into our equation \(x + 2y = 0\) we get \(1 + 2y = 0\), from which \(y = -1/2\). So a checkpoint is at the point (1,-0.5).

Step 4: Graph the equation

Plot the points (0,0) and (1,-0.5) and draw a straight line through these points to represent the equation \(x + 2y = 0\).

Key Concepts

Graphing Techniques

Graphing is a fundamental skill in algebra where visual representation helps make sense of equations. In this scenario, we use the graphing technique to illustrate the linear equation \(x + 2y = 0\).

To graph a line using intercepts, identify where the line crosses the axes. These points are called intercepts, which act as anchors for your line.

• The x-intercept is found by setting \(y = 0\) in the equation and solving for \(x\).
• The y-intercept is found by setting \(x = 0\) and solving for \(y\).

After determining the intercepts, plot them on a graph and draw a line through these points. Remember, any linear equation, such as \(x + 2y = 0\), will produce a straight line. This line extends infinitely in both directions along the coordinate plane.

To ensure accuracy when graphing, use a "checkpoint" — another point that you calculate by choosing a value for \(x\) or \(y\) and solving for the other variable. The checkpoint confirms the line's correct path through the plotted intercepts.

Intercepts in Algebra

Intercepts are crucial in understanding the behavior of linear equations on a graph. Let's explore how this applies to the equation \(x + 2y = 0\).

The x-intercept is where the graph crosses the x-axis, meaning \(y = 0\). Substitute \(y = 0\) into the equation to find the x-intercept. Here, substituting gives: \(x + 2(0) = 0\), resulting in \(x = 0\). Therefore, the graph passes through the point \((0, 0)\).

Similarly, the y-intercept is where the graph crosses the y-axis, meaning \(x = 0\). Plug \(x = 0\) into the equation to find this point. For the equation \(x + 2y = 0\), substituting \(x = 0\) yields: \(2y = 0\), leading to \(y = 0\). Hence, the y-intercept is also at \((0, 0)\).

In this particular equation, both intercepts are at the origin \((0, 0)\). This is a unique situation where the line passes directly through this single point, meaning the line essentially starts at the origin.

Coordinate Plane

The coordinate plane is a two-dimensional space formed by two perpendicular number lines: the x-axis and the y-axis. This plane is essential for plotting equations like \(x + 2y = 0\). Here’s how:

• X-axis: This is the horizontal line. It represents values of \(x\).
• Y-axis: This is the vertical line. It measures values of \(y\).

Together, these axes intersect at the "origin," which is the point \((0, 0)\). The entire plane is divided into four quadrants by these axes, each representing different signs of \(x\) and \(y\).

To graph the linear equation, you plot points on this grid, such as the intercepts \((0, 0)\) and the checkpoint \((1, -0.5)\). By drawing a straight line through these points, you reveal the solution set of the equation. Remember, every point on this line represents a pair \((x, y)\) that satisfies the equation \(x + 2y = 0\).

Understanding the coordinate plane is key in algebra as it allows visualization of equations, helping clarify their solutions and relationships.