Problem 22 - Algebra 1: Concepts and Skills

Question

Use the graphing method to tell how many solutions the system has. $$\begin{aligned} 3 x+2 y &=40 \\ -3 x-2 y &=8 \end{aligned}$$

Step-by-Step Solution

Answer

The system of equations has no solutions since the lines are parallel.

Step 1: Rewrite in Slope-Intercept Form

The equations need to be re-written in slope-intercept form, i.e., \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In order to do this, isolate \( y \) in both equations. For the first equation, it will look like this: \( y = -1.5x + 20 \). For the second equation, it will look like this: \( y = -1.5x - 4 \)

Step 2: Analyze The Slopes and Y-Intercept

The slope of both lines is \( -1.5 \), meaning the lines are parallel. The y-intercepts are \( 20 \) and \( -4 \) respectively, meaning the lines do not meet.

Step 3: Graph The Equations

Graph the each of the equations on the same graph. Since the lines are parallel, it is clear that the lines never meet and therefore do not have a common solution.

Step 4: Conclusion

Since the lines are parallel and do not intersect, we can conclude there are no solutions to the system of equations.

Key Concepts

graphing method

The graphing method is a visual way to find solutions to a system of equations. By graphing each equation on the same coordinate plane, we can see where the graphs intersect. This technique is often used to solve linear systems of two equations with two variables.

Using this method, the intersection point(s) of the lines represent the solution(s) to the system.

• If the lines intersect at a single point, there is one unique solution.
• If the lines are the same, they overlap completely, resulting in infinitely many solutions.
• If the lines are parallel, they do not intersect, indicating that there are no solutions.

Understanding the visual outcome when graphing lines can provide insights into the nature of the system's solutions.

slope-intercept form

The slope-intercept form is a way to express linear equations that makes graphing straightforward. The standard formula is given by:\[ y = mx + b \]where:

• \( m \) represents the slope of the line, which shows the steepness or angle of direction
• \( b \) is the y-intercept, the point where the line crosses the y-axis

By rewriting equations in this form, it becomes much easier to plot them on a graph. You simply identify the y-intercept \( b \) and use the slope \( m \) to find other points on the line. For instance, a slope of -1.5 indicates that the line decreases by 1.5 units vertically for every 1 unit it moves horizontally. The slope-intercept form simplifies the process of graphing and comparing lines.

parallel lines

Parallel lines are lines in a plane that never meet. They always maintain the same distance apart, regardless of how far they are extended. In terms of linear equations, parallel lines occur when two equations have the same slope but different y-intercepts.

For example, if two lines are represented in the slope-intercept form as \( y = -1.5x + 20 \) and \( y = -1.5x - 4 \), both have an identical slope of -1.5. Because of this same slope:

• Both lines run in the same direction and at the same angle.
• Since they have different y-intercepts (20 and -4, respectively), they do not touch at any point on the graph.

Understanding parallel lines is crucial in identifying systems of equations with no solutions when graphing.

no solutions

When tackling systems of equations, it's possible to find scenarios where no solutions exist. This happens when the equations describe parallel lines, which as we've discussed, never intersect.

Once the equations are put into the slope-intercept form, identical slopes with different y-intercepts indicate parallel lines. Therefore, these lines never have a meeting point, meaning no shared solution satisfies both equations simultaneously.

In our example, the lines defined by \( y = -1.5x + 20 \) and \( y = -1.5x - 4 \) don't intersect. This non-intersection visually confirms the absence of solutions. Recognizing what it means for lines to not meet helps in understanding when and why a system has no solutions.