Problem 54 - College Algebra with Modeling and Visualization

Question

Use the intersection-of-graphs method to solve the equation. Then solve symbolically. 2x = 3x - 1

Step-by-Step Solution

Answer

The solution to the equation is \(x = 1\).

Step 1: Interpret the Problem

We are given the equation \(2x = 3x - 1\). Our goal is to find the value of \(x\) that satisfies this equation using the intersection-of-graphs method and then solve it symbolically.

Step 2: Intersection of Graphs Method

To use the intersection-of-graphs method, we need to graph the equations \(y_1 = 2x\) and \(y_2 = 3x - 1\), and find their point of intersection. This point's x-coordinate will be the solution to the original equation.

Step 3: Graph the Equations

Plot the lines on the graph:- For \(y_1 = 2x\), plot points like (0,0), (1,2), and (2,4) and draw the line through them.- For \(y_2 = 3x - 1\), plot points like (0,-1), (1,2), and (2,5) and draw the line through them.

Step 4: Identify the Intersection Point

The lines \(y_1 = 2x\) and \(y_2 = 3x - 1\) intersect at the point where they have the same \(x\) and \(y\) values. By observation, the intersection occurs at point (1, 2), indicating that \(x = 1\) is the solution.

Step 5: Symbolic Solution

To solve symbolically, rearrange the original equation:\[2x = 3x - 1\]Subtract \(2x\) from both sides:\[0 = x - 1\]Add 1 to both sides:\[x = 1\]Thus, the solution is \(x = 1\).

Step 6: Verify the Solution

Substitute \(x = 1\) back into the original equation to verify:- Left side: \(2(1) = 2\)- Right side: \(3(1) - 1 = 2\)Both sides are equal, confirming that the solution \(x = 1\) is correct.

Key Concepts

Graph Intersection

When we use the graph intersection method to solve an equation, we're essentially looking for the point where two lines cross each other on a graph. This crossing point, called the intersection, has the same
x- and y-values for both lines, which means it satisfies both equations simultaneously.
To apply this approach, we need to:

• Express each side of the equation as a separate function. For example, with the equation \(2x = 3x - 1\), we can define two functions: \(y_1 = 2x\) and \(y_2 = 3x - 1\).
• Graph these functions on a coordinate plane. Each function will appear as a line.
• Determine the point where the two lines intersect. This point's x-coordinate will provide the solution to the original equation.

In our example, the lines for \(y_1 = 2x\) and \(y_2 = 3x - 1\) intersect at the point \((1, 2)\). Hence, \(x = 1\) solves the equation. This graphical approach provides a visual confirmation of conceptual understanding.

Linear Equations

Linear equations represent equations where each term is either a constant or the product of a constant and a single variable. These equations, such as \(2x = 3x - 1\), graph as straight lines.
Linear equations can be written in various forms, but they most commonly appear in the slope-intercept form
\(y = mx + b\), where:

• \(m\) represents the slope of the line, indicating its steepness.
• \(b\) denotes the y-intercept, which is the point where the line crosses the y-axis.

In our exercise, converting \(2x = 3x - 1\) into the standard linear forms - \(y_1 = 2x\) and \(y_2 = 3x - 1\) - helps in visualizing them as lines on a graph. By doing so, we can understand better how their intersection can reveal the solution of these equations. This method is particularly effective for simple linear equations, providing a clear illustration of the solution process.

Symbolic Solution

A symbolic solution involves manipulating the equation using algebra to find the value of the variable. This approach is systematic and focuses on isolating the variable on one side to get a straightforward solution.
Let's outline the steps again using \(2x = 3x - 1\):

• Start by simplifying the equation to isolate the variable, \(x\). Here, we subtract \(2x\) from both sides: \(0 = x - 1\).
• Then, we solve for \(x\) by adding 1 to both sides: \(x = 1\).
• It's crucial to verify the solution by substituting \(x = 1\) back into the original equation to ensure both sides are equal.

The solution \(x = 1\) checks out because substituting back results in equality: \(2(1) = 2\) equals \(3(1) - 1 = 2\). Symbolic solutions offer a quick, algebraic way to reach the answer, illustrating the precision and efficiency of algebraic methods.