Question

the first line in the system of equations is graphed on the coordinate plane. graph the second line to find the solution to the system.\\(x + 2y = -2\\)\\(y = \frac{1}{2}x - 5\\)

Explanation:

Step1: Rewrite first equation in slope - intercept form

We have the equation \(x + 2y=-2\). Solve for \(y\):
Subtract \(x\) from both sides: \(2y=-x - 2\).
Divide each term by 2: \(y=-\frac{1}{2}x - 1\).
The slope of the first line is \(m_1 =-\frac{1}{2}\) and the \(y\) - intercept is \(b_1=-1\) (the point \((0, - 1)\)).

Step2: Analyze the second equation

The second equation is \(y = \frac{1}{2}x-5\). The slope of the second line is \(m_2=\frac{1}{2}\) and the \(y\) - intercept is \(b_2 = - 5\) (the point \((0,-5)\)).

Step3: Find the intersection point (solution)

To find the solution of the system, we can set the two equations equal to each other (since at the intersection point, the \(y\) - values are equal).
Set \(-\frac{1}{2}x - 1=\frac{1}{2}x-5\).
Add \(\frac{1}{2}x\) to both sides: \(-1=x - 5\).
Add 5 to both sides: \(x = 4\).
Substitute \(x = 4\) into \(y=\frac{1}{2}x-5\): \(y=\frac{1}{2}(4)-5=2 - 5=-3\).

Answer:

The solution to the system of equations is \(x = 4,y=-3\) or the ordered pair \((4,-3)\).