Problem 94 - 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. $$2 x+y=4$$

Step-by-Step Solution

Answer

The graph intersects the \(x\)-axis at \(x=2\) and the \(y\)-axis at \(y=4\).

Step 1: Solve for \(y\)

The first step is to solve the equation for \(y\). With the given equation \(2x + y = 4\), solving for \(y\) involves isolating \(y\) on one side of the equation. That's done by subtracting \(2x\) from both sides of the equation. So, \(y\) equals to \(4 - 2x\).

Step 2: Graph the equation

Input the obtained function for \(y\) into the graphing utility tool. By inputting \(y = 4 - 2x\), a straight line that is sloping downwards from left to right will be seen. This line represents all the possible ordered pairs \(x\) and \(y\) that could satisfy the equation.

Step 3: Identify the x-intercept(s)

The \(x\)-intercept of a graph is the value of \(x\) where \(y = 0\). Looking at the graph, this is the point where the line crosses the \(x\)-axis. From the equation \(y = 4 - 2x\), when \(y = 0\), \(x\) will equal \(2\), which is the \(x\)-intercept.

Step 4: Identify the y-intercept(s)

The \(y\)-intercept of a graph is the value of \(y\) where \(x = 0\). Looking at the graph, this is the point where the line crosses the \(y\)-axis. From the equation \(y = 4 - 2x\), when \(x = 0\), \(y\) will equal \(4\), which is the \(y\)-intercept.

Key Concepts

Solving for y

To start graphing a linear equation, we need to solve it for \(y\), which means isolating \(y\) on one side. This helps in understanding how \(y\) changes with different \(x\) values. For the equation \(2x + y = 4\), solve for \(y\) by subtracting \(2x\) from both sides. This simple manipulation leaves you with \(y = 4 - 2x\). Similarly, for any linear equation of the form \(Ax + By = C\), you would isolate \(y\) by rearranging the terms: move \(Ax\) to the other side and divide by \(B\). This results in equation of the form \(y = mx + c\), where \(m\) represents the slope and \(c\) the \(y\)-intercept.

x-intercept

Finding the \(x\)-intercept means determining the point where the graph crosses the \(x\)-axis. At this point, \(y\) is always \(0\). To find the \(x\)-intercept from the equation \(y = 4 - 2x\), set \(y\) to \(0\) and solve for \(x\). You will get:

• \(0 = 4 - 2x\)
• Rearranging gives \(2x = 4\)
• So, \(x = 2\)

Thus, the \(x\)-intercept is \((2, 0)\). Every time you want to find the \(x\)-intercept for any line, just make \(y=0\) and solve for \(x\). Simple!

y-intercept

The \(y\)-intercept is the point where the line crosses the \(y\)-axis. At this point, \(x\) is always \(0\). For our equation, \(y = 4 - 2x\), substitute \(x = 0\):

• \(y = 4 - 2 \cdot 0\)
• \(y = 4\)

Hence, the \(y\)-intercept is \((0, 4)\). For any linear equation, just substitute \(x=0\) to find the \(y\)-intercept. This point will help you quickly place the line on a graph.

Graphing utilities

Graphing utilities, such as graphing calculators or software, make plotting linear equations much easier. After solving the equation for \(y\), simply input the equation into the graphing utility. In this exercise, you would enter \(y = 4 - 2x\). The graph displayed should be a straight line because it's a linear equation. These tools often have features to help you identify intercepts directly from the graph by showing where the line crosses the axes. Using them can save time and reduce errors, making it easier to visualize the equation's behavior. Try exploring options and features available in these utilities to better utilize them for checking solutions and understanding results.