Problem 36 - Intermediate Algebra

Question

Solve each equation graphically. See Example \(6 .\) $$ 2 x-3=3 x-3 $$

Step-by-Step Solution

Answer

The equation has no solution as the lines are parallel, so they don't intersect.

Step 1: Understand the Equation

The equation we need to solve graphically is \(2x - 3 = 3x - 3\). Solving it graphically means we'll find where the two expressions, \(y_1 = 2x - 3\) and \(y_2 = 3x - 3\), intersect when plotted on a graph.

Step 2: Rearrange the Equation

To graph it, split the equation into two separate expressions: \(y_1 = 2x - 3\) and \(y_2 = 3x - 3\). We will plot these two lines on the same set of axes.

Step 3: Choose and Plot Values for y1

Choose values for \(x\) and calculate corresponding values for \(y_1 = 2x - 3\). For example, if \(x = 0\), then \(y_1 = -3\); if \(x = 1\), then \(y_1 = -1\). Plot these points on a graph.

Step 4: Choose and Plot Values for y2

Choose values for \(x\) and calculate corresponding values for \(y_2 = 3x - 3\). For example, if \(x = 0\), then \(y_2 = -3\); if \(x = 1\), then \(y_2 = 0\). Plot these points on the same graph as \(y_1\).

Step 5: Graph Intersections

Observe the graph where both lines \(y_1\) and \(y_2\) are plotted. Find the point where they intersect. This intersection's \(x\)-value is the solution to the equation.

Step 6: Analyze the Intersection

As you review the graph, you’ll see that the lines \(y_1 = 2x - 3\) and \(y_2 = 3x - 3\) are actually parallel. Since parallel lines do not intersect, there is no solution to the equation when graphed.

Key Concepts

Parallel Lines

Parallel lines are lines in a plane that never meet; they are always the same distance apart. In other words, no matter how far you extend them, they will never cross each other. This is a crucial concept when dealing with graphical solutions of equations. If two lines are parallel, they represent equations without a common solution, as they do not intersect at any point.

When you compare two linear equations to check for parallelism, observe their slopes. If the slopes are equal, then the lines are parallel. This can often be determined by the coefficients of the variable term 'x' when equations are written in the slope-intercept form, which is given as \(y = mx + c\). Here, \(m\) denotes the slope. If two equations have the same \(m\) value, they are parallel.

• Example: The lines \(y_1 = 2x - 3\) and \(y_2 = 3x - 3\) are parallel because they have different slopes, so originally they cannot be parallel. Make sure you actually check the equations' slope first for the correct assessment.

Intersection Point

The intersection point is where two lines on a graph meet or cross each other. It is the solution to the system of equations represented by those lines. For finding an intersection point, you graph both equations and look for a common point where they overlap.

This point provides the \(x\) and \(y\) values that satisfy both equations simultaneously. To identify this intersection correctly:

• Convert each equation to slope-intercept form, \(y = mx + b\), if needed.
• Plot each line on the coordinate plane.
• Find the exact spot where the two graphed lines touch.

In some cases, as with our example \(2x - 3 = 3x - 3\), there might be no intersection because the lines are parallel, as discussed earlier, leading to no solution.

No Solution

When you have parallel lines in a graph, they represent a system of equations that has no solution. What this means is that the lines do not intersect at any point, therefore, there is no set of \(x\) and \(y\) values that satisfy both equations.

This concept is important in understanding why some equations can't be solved graphically. In algebra, it signifies that the system is inconsistent. Whenever you find that two lines are parallel, it is an indication that your equations have no solution. This often happens when the lines have identical or equivalent slopes but different y-intercepts.

• In the exercise \(2x - 3 = 3x - 3\), we find that they appear to be parallel after checking the slope components, indicating no solution is possible graphically.

Plotting Graphs

Plotting graphs is the process of drawing lines on a coordinate plane to represent equations. It's an essential tool for visualizing and solving equations. To plot a graph, convert your equation into its simplest linear form, usually \(y = mx + b\), allowing you to understand the slope and y-intercept easily.

Once your equations are ready:

• Choose values for \(x\) and calculate the corresponding \(y\) values.
• Mark these points on a graph paper or digital platform.
• Draw the line that passes through these plotted points.

This approach helps visualize solutions or understand why there might not be a solution. For instance, in the equation \(2x - 3 = 3x - 3\), plotting both expressions shows that the lines are parallel, emphasizing the importance of this method in identifying solutions graphically.