Problem 59 - College Algebra with Modeling and Visualization

Question

Exercises \(59-66:\) Solve the linear equation with the intersection-of-graphs method. Approximate the solution to the nearest thousandth whenever appropriate. $$ 5 x-1.5=5 $$

Step-by-Step Solution

Answer

The solution is \(x = 1.3\).

Step 1: Rearrange the Equation

Start with the linear equation \(5x - 1.5 = 5\). The goal is to express this as two separate functions to use the intersection-of-graphs method. We leave 5 on the right, and rearrange the left: \(5x - 1.5\) becomes \(y_1 = 5x - 1.5\) and \(y_2 = 5\).

Step 2: Set Up the Functions

Define \(y_1 = 5x - 1.5\) as the first function and \(y_2 = 5\) as the second function. These will be the equations of two lines on a graph.

Step 3: Graph the Functions

Graph \(y_1 = 5x - 1.5\) and \(y_2 = 5\) on the same set of axes. The line for \(y_2 = 5\) is horizontal because it is constant. The line for \(y_1 = 5x - 1.5\) has a slope of 5 and a y-intercept of -1.5.

Step 4: Find the Intersection

Look at the graph and find the point where the two lines intersect. The x-coordinate of this intersection point is the solution to the equation. Algebraically, set \(5x - 1.5\) equal to 5 and solve for \(x\): \(5x - 1.5 = 5 \rightarrow 5x = 6.5 \rightarrow x = 1.3\).

Step 5: Approximate the Solution

Since the solution must be to the nearest thousandth, ensure accuracy. The intersection point gives \(x = 1.3\), which is already at the nearest thousandth precision.

Key Concepts

Intersection of Graphs Method

The intersection of graphs method is a visual approach to finding the solution to a system of equations or a single equation presented as two separate functions. Instead of solving the equation algebraically, we graphically depict the problem on an x-y coordinate plane.

In this context, the linear equation given, which is **\(5x - 1.5 = 5\)**, can be represented graphically by splitting it up into two functions. You define:

• \(y_1 = 5x - 1.5\)
• \(y_2 = 5\)

By doing this, you literally create two lines on a graph. The method seeks to determine where these two lines intersect. The intersection point gives you the solution to the equation. It's where both functions have the same values of \(x\) and \(y\), fulfilling the equation's condition.

Once plotted, the \(x\)-coordinate of this intersection point tells us the solution to the original equation. This method is particularly beneficial because it provides a visual verification of the solution and clearly demonstrates the relationship between the functions.

Graphing Linear Functions

Graphing linear functions involves plotting straight lines on a coordinate grid, defined by the equation \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. In our example, the functions are \(y_1 = 5x - 1.5\) and \(y_2 = 5\).

The first function, \(y_1 = 5x - 1.5\), is a line with a slope \(m = 5\) and a y-intercept of \(-1.5\). This tells you that for every unit the \(x\) value increases, the \(y\) value increases by 5. Start plotting this graph from the y-intercept \(-1.5\), and use the slope to determine the line's direction and steepness.

The second function, \(y_2 = 5\), is a horizontal line. It does not depend on \(x\) and remains constant at all points where \(y = 5\), no matter what \(x\) is. Plot this line across the graph.

Together, these steps create a visual of the linear functions and their behaviors, allowing us to clearly discern where they intersect.

Solving for x

Solving for \(x\) is the process of finding the value of the variable that makes the equation true. In our example, once visualized on the graph, the point where \(y_1\) and \(y_2\) intersect gives the solution for \(x\).

To do this algebraically, we set the expressions for \(y_1\) and \(y_2\) equal: \(5x - 1.5 = 5\).

Here's how we solve step by step:

• Add 1.5 to both sides to isolate terms with \(x\): \(5x = 6.5\)
• Divide both sides by 5 to solve for \(x\): \(x = 1.3\)

This calculation shows that the intersection of \(y_1\) and \(y_2\) occurs at \(x = 1.3\). Looking at a graph offers visual confirmation, but this arithmetic ensures precision and confirms that the intersection point accurately solves the equation to the nearest thousandth.