Problem 97 - Introductory Algebra for College Students

Question

Use a graphing utility to graph each equation. You will need to solve the equation for \(y\) before entering it. Use the graph displayed on the screen to identify the \(x\)-intercept and the \(y\)-intercept. \(4 x-2 y=-40\)

Step-by-Step Solution

Answer

The equation of the line is \(y = 2x + 20\), the x-intercept is (-10, 0), and the y-intercept is (0, 20).

Step 1: Rearrange the Equation

The given equation is \(4x - 2y = -40\). Solve this equation for \(y\) to be able to plot it on a graph. Rearranging terms, divide every term in the equation by 2 to get \(2x - y = -20\). Add \(y\) to both sides and subtract -20 from both sides to isolate \(y\). So the equation becomes \(y = 2x + 20\)

Step 2: Graphing the Equation

Plot the equation \(y = 2x + 20\). Start with the y-intercept (0, 20) since when \(x = 0, y = 20\), and then for each +1 change in \(x\), \(y\) will increase by 2. Mark these points on the plane and connect them to form the line.

Step 3: Identifying the Intercepts

The x-intercept can be found by setting \(y = 0\) in the equation and solving for \(x\). Placing \(y = 0\) in the equation \(y = 2x + 20\), we get \(0 = 2x + 20\). Solving for \(x\) gives us \(x = -10\). Therefore, the x-intercept is -10, or the point (-10, 0). The y-intercept, as calculated before is 20 or the point (0, 20).

Key Concepts

Understanding the x-intercept

The x-intercept of a line on a graph is the point where the line crosses the x-axis. This point is of particular interest because it tells us the value of \(x\) when \(y\) equals zero. To find the x-intercept from an equation, you replace \(y\) with zero and solve for \(x\).

For the equation \(y = 2x + 20\), setting \(y = 0\) gives us:

• \(0 = 2x + 20\)
• Subtract 20 from both sides: \(-20 = 2x\)
• Divide both sides by 2: \(x = -10\)

So, the x-intercept is the point \((-10, 0)\). This means the line will intersect the x-axis at \(x = -10\). To visualize it on the graph, locate the point where the line meets the x-axis, which should be exactly at this position.

Understanding the y-intercept

The y-intercept of a line is where the line crosses the y-axis. This is the point where \(x\) is zero, often the starting point when graphing a linear equation. Finding the y-intercept is straightforward; set \(x = 0\) in the equation and solve for \(y\).

Given the equation \(y = 2x + 20\), when \(x = 0\):

• \(y = 2(0) + 20\)
• \(y = 20\)

Thus, the y-intercept is \((0, 20)\). On the graph, this is where the line will meet the y-axis, at \(y = 20\). It serves as a vital reference point in plotting the graph. Consider this as the starting point when drawing the straight line representing the equation.

Solving for y in linear equations

Solving an equation for \(y\) means rearranging the equation so that \(y\) is by itself on one side of the equation. This is a crucial step for graphing because it allows you to more easily identify both the slope and intercepts.

Let's take the equation \(4x - 2y = -40\) and solve it for \(y\):

• Subtract \(4x\) from each side: \(-2y = -4x - 40\)
• Divide every term by \(-2\): \(y = 2x + 20\)

Now, the equation \(y = 2x + 20\) is in the slope-intercept form, \(y = mx + b\). This form makes it simple to graph because you can instantly see the slope, \(m = 2\), and the y-intercept, \(b = 20\). Understanding this process of solving for \(y\) is critical for efficiently and correctly plotting linear graphs.