Problem 98 - College Algebra

Question

Explain how to use intercepts to graph the general form oE a line's equation.

Step-by-Step Solution

Answer

In order to graph the general form of a line's equation using intercepts, first calculate the x-intercept by setting y equal to zero in the equation and solving for x. Then, calculate the y-intercept by setting x to zero and solving for y. These two intercepts can then be plotted on the graph as points, and a line can be drawn through the two points to form the line graph of the equation.

Step 1: Find the X-Intercept

Set 'y' equal to zero in the equation and solve for 'x'. This will give you the x-coordinate of the x-intercept. If the equation is in the form Ax+By=C, then the x-intercept is \( \frac{C}{A} \) when B=0.

Step 2: Find the Y-Intercept

Set 'x' to zero and solve for 'y'. This will give the y-coordinate of the y-intercept. If the equation is in the form Ax+By=C, then the y-intercept is \( \frac{C}{B} \) when A=0.

Step 3: Plot the intercepts

Plot both the x-intercept and the y-intercept points on the graph.

Step 4: Draw a straight line through the intercepts

After both intercepts are plotted on the graph, draw a straight line that passes through both points. This line represents the line's equation, and all points on this line are solutions to the equation.

Key Concepts

X-Intercept

Understanding the x-intercept of a line is a fundamental aspect in graphing linear equations. It is the point where the line crosses the x-axis, which means it is where the value of y is zero. When graphing, you can find the x-intercept by setting the y-variable to zero and solving for x.

For example, in an equation of the form Ax + By = C, if you set y to 0, the equation simplifies to Ax = C. To find the x-intercept, simply solve for x, which gives you \( x = \frac{C}{A} \). It’s essential to note that if A is 0, the line is horizontal and doesn't have an x-intercept unless C is also 0, in which case, the line lies on the x-axis itself.

Y-Intercept

The y-intercept is the point where the line intersects the y-axis. At this point, the value of x is zero. To find the y-intercept when you have an equation in general form, such as Ax + By = C, set x to zero and solve for y.

This will give you \( y = \frac{C}{B} \). This step is crucial as it helps you determine one of the two points needed to graph the line. Remember, if B is 0, the line is vertical and does not have a y-intercept, except when C is zero; then, it coincides with the y-axis as well.

General Form of a Line

The general form of a line's equation is usually written as Ax + By = C, where A, B, and C are constants. This form is particularly useful when graphing because it allows for a straightforward identification of the intercepts upon rearranging for either variable. The coefficient A controls the slope in conjunction with B, and C determines the distance from the origin.

Interpreting the General Form

The coefficients A and B convey information about the slope: if B is zero, the line is vertical, and if A is zero, the line is horizontal. C reveals the intercept that doesn't involve setting the other variable to zero. When A or B is not zero, you have a sloped line, and both x-intercept and y-intercept can be found using the previously mentioned steps.

Plotting Intercepts

Plotting intercepts is a simple yet powerful method for graphing lines. After determining the x-intercept and y-intercept, these points provide a frame of reference to accurately draw the line on a graph.

First, find and mark the x-intercept on the x-axis. Next, locate and plot the y-intercept on the y-axis. Drawing a straight line passing through these two points will give you the graph of the equation. It's vital to ensure that the line extends through the entire graph to accurately depict all solutions of the equation. This method works because linear equations produce straight lines, so two distinct points are all that's needed. By utilizing intercepts, you create the most clear and precise representation of the line.