Question

solve the following system of equations graphically on the set of axes below.
$y = \frac{1}{2}x + 6$
$x + y = 3$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Analyze the first equation

The first equation is \( y = \frac{1}{2}x + 6 \), which is in slope - intercept form (\(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept). The slope \(m=\frac{1}{2}\) and the y - intercept \(b = 6\). So we can start by plotting the y - intercept at \((0,6)\). Then, using the slope (rise over run), from \((0,6)\), we can go up 1 unit and right 2 units to get the next point \((2,7)\), or down 1 unit and left 2 units to get \((- 2,5)\).

Step2: Analyze the second equation

The second equation is \(x + y=3\). We can rewrite it in slope - intercept form by solving for \(y\): \(y=-x + 3\). Here, the slope \(m=- 1\) and the y - intercept \(b = 3\). We plot the y - intercept at \((0,3)\). Then, using the slope, from \((0,3)\), we can go down 1 unit and right 1 unit to get \((1,2)\), or up 1 unit and left 1 unit to get \((-1,4)\).

Step3: Find the intersection point

After plotting both lines, we look for the point where they intersect. To find the intersection algebraically (to confirm), we can set the two equations equal to each other:
\(\frac{1}{2}x+6=-x + 3\)
Add \(x\) to both sides: \(\frac{1}{2}x+x+6=-x+x + 3\), which simplifies to \(\frac{3}{2}x+6 = 3\)
Subtract 6 from both sides: \(\frac{3}{2}x+6 - 6=3 - 6\), so \(\frac{3}{2}x=-3\)
Multiply both sides by \(\frac{2}{3}\): \(x=-3\times\frac{2}{3}=-2\)
Substitute \(x = - 2\) into \(y=-x + 3\): \(y=-(-2)+3=2 + 3=5\)
So the intersection point (the solution to the system) is \((-2,5)\).

Answer:

The solution to the system of equations is \(x=-2\), \(y = 5\) (or the point \((-2,5)\)).