Problem 5 - Algebra 1: Concepts and Skills

Question

Graph the system of linear equations. Does the system have exactly one solution, no solution, or infinitely many solutions? Explain. $$\begin{aligned} &-6 x+2 y=4\\\ &-9 x+3 y=12 \end{aligned}$$

Step-by-Step Solution

Answer

The system of equations has no solution, as the graphs of the two equations are parallel lines that do not intersect.

Step 1: Express equations in slope-intercept form

Firstly, express both equations in slope-intercept form which is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Given equations are \(-6x + 2y = 4\) and \(-9x + 3y = 12\). When expressed in slope - intercept form, they become \(y = 3x + 2\) and \(y = 3x + 4\) respectively.

Step 2: Graph the equations

Secondly, for each equation plot the y-intercept and then use the slope to plot more points on the line. Join the points to draw the line for each equation.

Step 3: Analyze the graph

After graphing, observe the lines. Both lines are parallel but not coincident. Thus, they do not intersect at any point.

Step 4: Conclusion

Since the lines do not intersect and are not coincident, it can be concluded that the system of equations has no solution.

Key Concepts

Slope-Intercept Form

When dealing with linear equations, one of the easiest and most common forms to work with is the slope-intercept form. This is expressed as \(y = mx + c\), where \(m\) represents the slope of the line and \(c\) the y-intercept, which is the point where the line crosses the y-axis.

In the context of our given problem, by rearranging the equations to match this form, we can immediately see some crucial characteristics of each line, such as the inclination (given by the slope) and where they meet the y-axis. This can offer us a direct insight into whether the lines will intersect, be parallel, or perhaps be the same line (coincident).

It’s important to note that the slope is a measure of how steep a line is, and the y-intercept is a constant that indicates the precise location on the y-axis where the line rests when \(x=0\). Thus, the slope-intercept form not only simplifies the process of graphing but also facilitates the understanding of the line’s behavior.

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra that provides a visual representation of the equation. The process starts by identifying the y-intercept point on the graph, which can be plotted directly from the value of \(c\) in the slope-intercept form of the equation.

After plotting this point, the slope \(m\) tells us how to move from the y-intercept to the next point: up or down by \(m\) units in the y-direction for each one unit we move to the right in the x-direction. This 'rise over run' technique allows us to plot a second point. By connecting these two points with a straight line, we have graphed our equation.

Using this method, we can graph any linear equation quickly and accurately, ensuring that we have a clear visual tool to analyze the system of equations we're working with. This approach is exceptionally beneficial when working with systems of equations, as it aids in determining whether the lines intersect at a point, which would indicate a single solution.

Analyzing Graphs of Linear Systems

Analyzing graphs of linear systems is the final step after graphing the individual equations. This involves looking at where the lines lie in relation to each other. If the lines intersect at a single point, the system has one solution, which is the point of intersection.

However, if the lines are parallel, as showcased in our given problem, they will never intersect, resulting in no solutions. Parallel lines have the same slope but different y-intercepts. Another possibility is that the lines could be coincident, meaning they are essentially the same line; this would result in infinitely many solutions as every point on the line satisfies both equations.

A careful evaluation of the graph, noting the slopes and y-intercepts, can therefore determine the solution set of the system. Since our example equations resulted in parallel lines, the conclusion is that the system does not have a solution. Graphical analysis is a powerful tool that confirms the characteristics of systems of equations and helps to visualize the nature of their solutions.