Question

solve each system of equations by graphing. clearly identify yo
1.
\

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

2.
\

$$\begin{cases} x - y = 7\\\\ x - y = -4 \\end{cases}$$

Explanation:

Step1: Solve the first system

We have the system of equations:

$$
LATEXBLOCK0
$$

To find the solution, we set the two equations equal to each other since they both equal \(y\):

$$ \frac{2}{3}x - 1=-x + 4 $$

Add \(x\) to both sides:

$$ \frac{2}{3}x+x - 1=4 $$

Combine like terms: \(\frac{2}{3}x+\frac{3}{3}x=\frac{5}{3}x\), so we have \(\frac{5}{3}x - 1 = 4\)
Add 1 to both sides:

$$ \frac{5}{3}x=4 + 1=5 $$

Multiply both sides by \(\frac{3}{5}\):

$$ x = 5\times\frac{3}{5}=3 $$

Now substitute \(x = 3\) into \(y=-x + 4\):

$$ y=-3 + 4 = 1 $$

So the solution to the first system is \((3,1)\)

Step2: Solve the second system

We have the system of equations:

$$
LATEXBLOCK1
$$

Rewrite the equations in slope - intercept form (\(y=mx + b\)):
For \(x - y=7\), we get \(y=x - 7\)
For \(x - y=-4\), we get \(y=x + 4\)
The slopes of both lines are \(m = 1\) (since the coefficient of \(x\) is 1 in both equations), and the y - intercepts are \(-7\) and \(4\) respectively. Since the slopes are equal and the y - intercepts are different, the two lines are parallel and there is no solution.

Answer:

1. The solution of the system \(

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

\) is \((3,1)\)

1. The system \(

$$\begin{cases}x - y = 7\\x - y=-4\end{cases}$$

\) has no solution.