Question

systems of equations graphing method
graded assignment (show all work!!!) name:
date: period :
use graphing method to solve and name the type of solutions

1. \\(\

$$\begin{cases} x = -4 \\\\ 3x + 4y = 8 \\end{cases}$$

\\)
solution:

1. \\(\

$$\begin{cases} y = 2x - 4 \\\\ 6x - 3y = -3 \\end{cases}$$

\\)
solution:

1. \\(\

$$\begin{cases} y = x - 4 \\\\ y = x - 4 \\end{cases}$$

\\)
solution:

1. \\(\

$$\begin{cases} y = \\frac{1}{2}x + 2 \\\\ y = -\\frac{3}{4}x - 3 \\end{cases}$$

\\)
solution:

Explanation:

Problem 1

Step1: Analyze the first equation

The first equation is \( x = -4 \), which is a vertical line passing through \( x = -4 \) on the x - axis.

Step2: Substitute \( x=-4 \) into the second equation

The second equation is \( 3x + 4y=8 \). Substitute \( x = - 4\) into it:
\[

$$\begin{align*} 3\times(-4)+4y&=8\\ -12 + 4y&=8\\ 4y&=8 + 12\\ 4y&=20\\ y&=5 \end{align*}$$

\]

Step3: Graph the lines

The line \( x=-4 \) is vertical. The line \( 3x + 4y = 8\) can be rewritten in slope - intercept form (\(y=mx + b\)) as \(y=-\frac{3}{4}x + 2\) (by solving \(3x + 4y=8\) for \(y\): \(4y=-3x + 8\), \(y =-\frac{3}{4}x+2\)). The two lines intersect at the point \((-4,5)\). Since the two lines intersect at exactly one point, the system has one unique solution.

Step1: Analyze the equations

The system is \(

$$\begin{cases}y = 2x-4\\6x-3y=-3\end{cases}$$

\). Let's rewrite the second equation in slope - intercept form. Solve \(6x-3y=-3\) for \(y\):
\[

$$\begin{align*} -3y&=-6x - 3\\ y&=2x + 1 \end{align*}$$

\]

Step2: Compare the slopes and y - intercepts

The first equation \(y = 2x-4\) has a slope \(m_1 = 2\) and y - intercept \(b_1=-4\). The second equation \(y = 2x + 1\) has a slope \(m_2=2\) and y - intercept \(b_2 = 1\). Since the slopes are equal (\(m_1=m_2\)) and the y - intercepts are different (\(b_1
eq b_2\)), the two lines are parallel. Parallel lines never intersect, so the system has no solution.

Step1: Analyze the equations

The system is \(

$$\begin{cases}y=x - 4\\y=x - 4\end{cases}$$

\). Both equations are the same. This means that the two lines are coincident (they lie on top of each other).

Step2: Determine the number of solutions

When two lines are coincident, they intersect at infinitely many points. So the system has infinitely many solutions.

Answer:

\((-4,5)\) (one unique solution)

Problem 2