Questions: Question 4 Systems of Equations Below are three systems of equations. Examine the slope and vertical intercepts of each to determine the type of solutions to that system of equations y1 = 2x - 4 y2 = -2x + 4 Select an answer y1 = 2.5x + 4 y2 = 5/2 x + 4 y1 = 2x + 4 y2 = 2x + 2

Question 4

Systems of Equations Below are three systems of equations. Examine the slope and vertical intercepts of each to determine the type of solutions to that system of equations

y1 = 2x - 4
y2 = -2x + 4

Select an answer

y1 = 2.5x + 4
y2 = 5/2 x + 4

y1 = 2x + 4
y2 = 2x + 2

Transcript text: Question 4 Systems of Equations Below are three systems of equations. Examine the slope and vertical intercepts of each to determine the type of solutions to that system of equations \[ \begin{array}{c} y_{1}=2 x-4 \\ y_{2}=-2 x+4 \end{array} \] Select an answer \[ \begin{array}{l} y_{1}=2.5 x+4 \\ y_{2}=\frac{5}{2} x+4 \end{array} \] \[ \begin{array}{l} y_{1}=2 x+4 \\ y_{2}=2 x+2 \end{array} \]

Solution

Solution Steps

To determine the type of solutions for each system of equations, we need to compare the slopes and y-intercepts of the lines. If the slopes are equal and the y-intercepts are different, the lines are parallel and there is no solution. If the slopes are equal and the y-intercepts are the same, the lines are coincident and there are infinitely many solutions. If the slopes are different, the lines intersect at one point, and there is exactly one solution.

Step 1: Analyze the First System of Equations

The first system of equations is: \[ \begin{align_} y_1 &= 2x - 4 \\ y_2 &= -2x + 4 \end{align_} \] The slopes are \(2\) and \(-2\), which are different. Therefore, the lines intersect at one point, indicating there is exactly one solution.

Step 2: Analyze the Second System of Equations

The second system of equations is: \[ \begin{align_} y_1 &= 2.5x + 4 \\ y_2 &= \frac{5}{2}x + 4 \end{align_} \] The slopes are \(2.5\) and \(\frac{5}{2}\), which are equal. The y-intercepts are both \(4\). Since both the slopes and y-intercepts are the same, the lines are coincident, indicating there are infinitely many solutions.

Step 3: Analyze the Third System of Equations

The third system of equations is: \[ \begin{align_} y_1 &= 2x + 4 \\ y_2 &= 2x + 2 \end{align_} \] The slopes are both \(2\), but the y-intercepts are \(4\) and \(2\), which are different. Therefore, the lines are parallel and do not intersect, indicating there is no solution.