Question

solve the system of linear equations by graphing.
$y = \frac{1}{2}x + 4$
$y = -3x - 3$
the solution is (□, □).

Explanation:

Step1: Analyze the first equation

The equation \(y = \frac{1}{2}x + 4\) is in slope - intercept form (\(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept). The slope \(m_1=\frac{1}{2}\) and the y - intercept \(b_1 = 4\). To graph this line, we start by plotting the y - intercept at \((0,4)\). Then, using the slope \(\frac{1}{2}\) (which means for every 2 units we move to the right along the x - axis, we move up 1 unit along the y - axis), we can find another point. For example, when \(x = 2\), \(y=\frac{1}{2}(2)+4=1 + 4=5\), so the point \((2,5)\) is also on this line.

Step2: Analyze the second equation

The equation \(y=-3x - 3\) is also in slope - intercept form. The slope \(m_2=-3\) and the y - intercept \(b_2=-3\). We plot the y - intercept at \((0, - 3)\). Using the slope \(-3\) (which means for every 1 unit we move to the right along the x - axis, we move down 3 units along the y - axis), when \(x = 1\), \(y=-3(1)-3=-6\), so the point \((1, - 6)\) is on this line.

Step3: Find the intersection point

To solve the system of equations \(

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

\), we can set the two equations equal to each other since they both equal \(y\):
\(\frac{1}{2}x + 4=-3x-3\)
Multiply through by 2 to clear the fraction:
\(x + 8=-6x-6\)
Add \(6x\) to both sides:
\(x+6x + 8=-6x + 6x-6\)
\(7x+8=-6\)
Subtract 8 from both sides:
\(7x+8 - 8=-6 - 8\)
\(7x=-14\)
Divide both sides by 7:
\(x=-2\)

Step4: Find the value of y

Substitute \(x = - 2\) into one of the original equations. Let's use \(y=-3x - 3\).
\(y=-3(-2)-3=6 - 3=3\)

We can also verify by substituting \(x=-2\) into \(y=\frac{1}{2}x + 4\): \(y=\frac{1}{2}(-2)+4=-1 + 4 = 3\).

The point of intersection of the two lines (the solution to the system) is \((-2,3)\). When we graph the two lines, they will intersect at the point \((-2,3)\).

Answer:

\((-2, 3)\)