Problem 20 - Algebra and Trigonometry

Question

\(15-20\) m Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$ \left\\{\begin{array}{c}{12 x+15 y=-18} \\ {2 x+\frac{5}{2} y=-3}\end{array}\right. $$

Step-by-Step Solution

Answer

The system has infinitely many solutions, as both lines are identical.

Step 1: Write in Slope-Intercept Form

First, convert each of the linear equations into slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.For the equation \(12x + 15y = -18\), solve for \(y\):\[15y = -12x - 18 \y = -\frac{12}{15}x - \frac{18}{15} = -\frac{4}{5}x - \frac{6}{5}\]For the second equation \(2x + \frac{5}{2}y = -3\), solve for \(y\):\[\frac{5}{2}y = -2x - 3 \y = -\frac{4}{5}x - \frac{6}{5}\]

Step 2: Examine the Equations

Notice that both equations have been converted to the same slope-intercept form: \(y = -\frac{4}{5}x - \frac{6}{5}\). This implies that these two lines are identical, suggesting infinitely many solutions since they overlap completely.

Step 3: Plot the Equations

Though graphing confirms our algebraic conclusion, here's how you would do it:Plot the equation \(y = -\frac{4}{5}x - \frac{6}{5}\) on a graph. Start by finding the y-intercept, which is \(-\frac{6}{5}\), and plot this point. The slope is \(-\frac{4}{5}\), meaning for every 5 units you move horizontally, you move 4 units downward. Draw the line.Since the second equation is identical, it would lie over the first one, confirming the result.

Step 4: Interpret the Graph

Since the two lines coincide perfectly, they have infinitely many solutions, meaning they intersect at every point along the lines. Thus, the system does not have a unique solution but rather shares the solution set in its entirety, as both lines are the same.

Key Concepts

Slope-Intercept Form

In mathematics, understanding the slope-intercept form of a linear equation is essential. The slope-intercept form is given by the equation \(y = mx + b\). Here, \(m\) represents the slope of the line, which indicates how steep the line is. On the other hand, \(b\) is the y-intercept, which tells us where the line crosses the y-axis. This form makes it easy to quickly understand the characteristics of a line.

When converting a standard form equation like \(12x + 15y = -18\) to slope-intercept form, the goal is to solve for \(y\). We rearranged the equation to isolate \(y\) on one side, leading us to \(y = -\frac{4}{5}x - \frac{6}{5}\). This form allows us to easily identify the slope \(-\frac{4}{5}\) and the y-intercept \(-\frac{6}{5}\). Converting linear equations into the slope-intercept form helps us to visualize the line, especially when graphing.

Graphing Linear Equations

Graphing linear equations is a visual way to represent these equations and understand their solutions. Once the equation is in slope-intercept form, such as \(y = -\frac{4}{5}x - \frac{6}{5}\), drawing the line becomes straightforward.

• Start by plotting the y-intercept\(-\frac{6}{5}\) on the y-axis. This is the point where the line will cross the y-axis.
• Next, use the slope \(-\frac{4}{5}\) to determine the direction and steepness of the line. From the y-intercept, move down 4 units and 5 units to the right (which reflects the slope \(-4/5\)). Mark this point and draw a line through these points.

This method will give you a clear picture of the line. When graphing a system of equations, you plot each equation on the same graph to see how they interact. Whether the lines intersect at one point, are parallel, or coincide determines the number of solutions.

Infinitely Many Solutions

In systems of linear equations, sometimes the lines that represent the equations are identical. When this happens, the system of linear equations does not have a unique solution but instead has infinitely many solutions.

In our system, both equations \(y = -\frac{4}{5}x - \frac{6}{5}\) are identical after conversion. This identity shows that every point on one line is also a point on the other line. Thus, they overlap completely.

• When two lines have precisely the same slope and y-intercept, they are essentially the same line drawn on top of each other.
• This means the lines coincide, and thus, there are infinitely many solutions as every coordinate point on the line is a solution.

Understanding that infinitely many solutions mean the equations describe the same line is vital. This helps confirm the solution set is shared entirely, making it a crucial concept for mastering linear systems.