Problem 3 - Algebra and Trigonometry

Question

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{array}{l}{2 x-3 y=12} \\ {-x+\frac{3}{2} y=4}\end{array}\right.$$

Step-by-Step Solution

Answer

The system has no solution because the lines are parallel.

Step 1: Rewrite the Equations in Slope-Intercept Form

For each equation, solve for \(y\) in terms of \(x\) to get the slope-intercept form \(y = mx + b\). For the first equation, \(2x - 3y = 12\): \[ -3y = -2x + 12 \ y = \frac{2}{3}x - 4 \]For the second equation, \(-x + \frac{3}{2}y = 4\):\[ \frac{3}{2}y = x + 4 \ y = \frac{2}{3}x + \frac{8}{3} \]

Step 2: Graph the Equations

Graph each line on the same coordinate plane. - For \(y = \frac{2}{3}x - 4\), start at the y-intercept \((0, -4)\) and use the slope \(\frac{2}{3}\) (rise of 2 and run of 3) to plot the next points.- For \(y = \frac{2}{3}x + \frac{8}{3}\), start at the y-intercept \((0, \frac{8}{3})\) and similarly use a slope of \(\frac{2}{3}\). Draw both lines.

Step 3: Analyze the Graph

Observe the two lines to determine the number of solutions to the system. Since both lines have the same slope (\(\frac{2}{3}\)) but different y-intercepts (\(-4\) and \(\frac{8}{3}\)), the lines are parallel and do not intersect.

Step 4: Conclude the Solution Type

Since the lines are parallel and do not intersect, the linear system has no solutions. Parallel lines indicate that the equations have no points in common.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is essential for understanding and graphing linear systems. In this format, an equation appears as \[ y = mx + b \] where:

• \( m \) is the slope of the line
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Converting a linear equation to slope-intercept form involves solving for \( y \) in terms of \( x \). This helps to easily identify both the slope and y-intercept, which are crucial for graphing. In our exercise, the equations were initially in a different form but were converted into:

• \( y = \frac{2}{3}x - 4 \)
• \( y = \frac{2}{3}x + \frac{8}{3} \)

Both equations have the same slope (\( \frac{2}{3} \)), indicating that the lines are parallel.

Graphing Linear Equations

Graphing linear equations allows us to visually interpret solutions. Once in slope-intercept form, graphing these lines on a coordinate plane becomes straightforward. Start plotting by locating the y-intercept (\( b \)). For example:

• The first line starts at \( (0, -4) \).
• The second line starts at \( (0, \frac{8}{3}) \).

This starting point helps position the line on a graph. Next, use the slope \( m \), which is a ratio representing the rise over run (vertical change over horizontal change), to find another point on the line. For instance, a slope of \( \frac{2}{3} \) means going up two units and right three units from any point on the line. Connect these points to draw the complete line. Both lines are parallel in this exercise, leading to an important conclusion about the system of equations.

Systems of Equations

Understanding systems of equations involves interpreting multiple equations together, particularly their graphical representations. A system can have:

• One solution, where lines intersect at a single point.
• No solution, where lines are parallel and never intersect.
• Infinitely many solutions, where lines coincide completely.

Through graphing, you can quickly discern the type of solution a system possesses. In the exercise's system, both lines have the same slope but different y-intercepts, hence they are parallel. This shows there's no intersection point, giving us no solution. Parallel lines in the context of linear systems imply the equations are inconsistent, reflecting no common solution across the equations involved.