Problem 95 - Introductory and Intermediate Algebra for College Students 4th

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. $$3 x-y=9$$

Step-by-Step Solution

Answer

The x-intercept for the equation \(3x - y = 9\) is \(3\) and the y-intercept is \(-9\).

Step 1: Isolate y in the equation

The linear equation given is \(3x - y = 9\). In order to isolate \(y\), subtract \(3x\) from both sides of the equation. This yields the equation \(y = 3x - 9\).

Step 2: Graph the equation

Utilize a graphing tool to plot the equation \(y = 3x - 9\). This will result in a straight line, as it is a linear equation.

Step 3: Identify the intercepts

The x-intercept is the point where the line crosses the x-axis (i.e., where \(y = 0\)). For the given equation, substituting \(y = 0\) results in \(x = 3\). The y-intercept is the point where the line crosses the y-axis (i.e., where \(x = 0\)). For our equation, substituting \(x = 0\) results in \(y = -9\). Therefore, the x-intercept is \(3\) and the y-intercept is \(-9\).

Key Concepts

Solving for y

In algebra, solving for a variable, such as y, is a fundamental skill. When we're given an equation like 3x - y = 9, we want to get y on one side of the equals sign by itself. This is known as isolating the variable. To achieve that, you can perform balanced operations on both sides of the equation.

• Subtract 3x from each side, getting -y = -3x + 9.
• To make y positive, multiply each side by -1, resulting in y = 3x - 9.

Now the equation is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. This form makes graphing much easier and leads us to our next concept.

Graphing Utility

A graphing utility is an invaluable tool for visualizing algebraic equations. After rearranging the given equation into the form suitable for graphing, for instance, y = 3x - 9, you can input this function into a graphing calculator or software.

These utilities plot the equation on a coordinate plane, showing how y changes with x. You'll see a visual representation of the equation, typically a line for linear equations, on the graph. Points where the line crosses the axes represent intercepts, which are key to understanding the behavior of the equation.

X-Intercept

The x-intercept of a graph is a point where the line crosses the x-axis. This happens when the value of y is zero. To find the x-intercept algebraically, replace y with zero and solve the equation for x.

• From y = 3x - 9, set y to 0.
• So, 0 = 3x - 9. Add 9 to both sides to get 9 = 3x.
• Divide both sides by 3 to find x = 3.

Thus, the x-intercept is the point (3, 0), which you can also confirm using the graphing utility.

Y-Intercept

Similarly, the y-intercept is where the graph crosses the y-axis, and x is zero. To find the y-intercept from an equation, simply set x to zero and solve for y.

• Starting with y = 3x - 9, let x = 0.
• Substitute and find y = -9.

This calculation demonstrates that the y-intercept is the point (0, -9). The y-intercept is particularly helpful because it gives us a starting point when plotting a linear equation by hand on a graph, as well as an indicator of where the graph will cross the y-axis.