Problem 30 - Algebra A Combined Function

Question

Solve each system of linear equations by graphing. See Examples 3 through \(6 .\) \(\left\\{\begin{array}{l}x-2 y=-6 \\ -2 x+4 y=12\end{array}\right.\)

Step-by-Step Solution

Answer

Infinitely many solutions.

Step 1: Rearrange Each Equation into Slope-Intercept Form

To graph each line, first convert each equation to the slope-intercept form, \(y = mx + b\). Start with the first equation, \(x - 2y = -6\): Add \(2y\) to both sides to get \(x = 2y - 6\). Then add 6 to both sides and divide by 2, yielding \(y = \frac{1}{2}x + 3\).For the second equation, \(-2x + 4y = 12\): Add \(2x\) to both sides to get \(4y = 2x + 12\). Then divide every term by 4, which simplifies to \(y = \frac{1}{2}x + 3\).

Step 2: Graph Both Equations

Now that we have both equations in slope-intercept form, graph them. Both equations \(y = \frac{1}{2}x + 3\) have the same slope (\(\frac{1}{2}\)) and y-intercept (3), so they represent the same line. Start by plotting the y-intercept (0,3) and then use the slope to find another point. Since the slope is \(\frac{1}{2}\), from the y-intercept, go up 1 unit and right 2 units to plot another point at (2,4).

Step 3: Analyze the Graph

With both lines graphed, observe that they coincide, meaning they are the same line. This indicates the system has infinitely many solutions because every point on the line satisfies both equations.

Key Concepts

graphing linear equations

Graphing linear equations involves plotting them on a coordinate plane. Each linear equation represents a straight line. In the equation form, such as \(ax + by = c\), you can graph by determining two critical points on the line: one usually being the y-intercept and the other found using the slope.
This provides a visual representation that helps in understanding the relationship between variables.
The benefit of graphing equations lies in its ability to reveal intersections visually, allowing one to easily see solutions such as points of intersection, or even to notice overlapping lines that suggest special cases like infinitely many solutions.

slope-intercept form

The slope-intercept form of a linear equation is one of the most common forms used in algebra, expressed as \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) is the y-intercept.
This form is particularly useful for graphing because it immediately gives you both the slope and the point where the line crosses the y-axis.

• The slope \(m\) indicates the steepness of the line, calculated as the "rise over run" or the change in y divided by the change in x.
• The y-intercept \(b\) is the point on the graph where the line crosses the y-axis (where \(x = 0\)).

Converting equations to this form simplifies the graphing process, making it easy to plot the starting point and use the slope to determine the direction and steepness of the line.

infinitely many solutions

In the context of systems of linear equations, having infinitely many solutions means that there is not just one unique solution, but rather a whole set of solutions that satisfy the equations.
This situation occurs when the two lines representing the equations coincide, lying exactly on top of each other in the graph.

• If both equations in a system simplify to the same line, every point on that line is a solution.
• This indicates that there is no single intersection point because they overlap perfectly.

When graphing, if you notice the lines completely overlap, meaning you have infinitely many solutions.

solving by graphing

Solving a system of equations by graphing involves plotting each equation on the same coordinate grid to find where they intersect.
This method is particularly advantageous for visual learners or when a rough estimate of the solution is sufficient. The steps usually include:

• Convert each equation to slope-intercept form to make them easier to graph.
• Identify the y-intercept and plot it on the graph.
• Use the slope to find another point and draw the line.
• Find the point where the lines intersect; this point is the solution of the system.

When you solve by graphing, you may encounter cases like having one solution (intersection point), no solutions (parallel lines), or infinitely many solutions (coinciding lines). This visualization helps in comprehensively understanding the nature of the system's solutions.