Question

solve the following system of equations graphically on the set of axes below.

$y = x + 2$

$y = -\frac{1}{3}x + 8$

plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Graph \( y = x + 2 \)

The equation \( y = x + 2 \) is in slope - intercept form (\( y=mx + b \)) where the slope \( m = 1 \) and the y - intercept \( b = 2 \). To graph this line, we start by plotting the y - intercept at the point \( (0,2) \). Then, using the slope (rise over run, since \( m = 1=\frac{1}{1} \)), we move 1 unit up and 1 unit to the right from \( (0,2) \) to get the point \( (1,3) \), 1 unit up and 1 unit to the right from \( (1,3) \) to get \( (2,4) \), and so on. We can also move 1 unit down and 1 unit to the left from \( (0,2) \) to get \( (- 1,1) \), \( (-2,0) \), etc.

Step2: Graph \( y=-\frac{1}{3}x + 8 \)

The equation \( y =-\frac{1}{3}x + 8 \) is also in slope - intercept form with slope \( m=-\frac{1}{3} \) and y - intercept \( b = 8 \). We start by plotting the y - intercept at the point \( (0,8) \). Then, using the slope (rise over run, \( m =-\frac{1}{3}=\frac{- 1}{3} \)), we move 1 unit down and 3 units to the right from \( (0,8) \) to get the point \( (3,7) \), 1 unit down and 3 units to the right from \( (3,7) \) to get \( (6,6) \), and so on. We can also move 1 unit up and 3 units to the left from \( (0,8) \) to get \( (-3,9) \), etc.

Step3: Find the intersection point

The solution to the system of equations is the point where the two lines intersect. To find this point algebraically (to confirm the graphical solution), we set the two equations equal to each other:

$$
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$$

Substitute \( x = 4.5 \) into \( y=x + 2 \), we get \( y=4.5+2=6.5=\frac{13}{2} \). So the intersection point (the solution of the system) is \( (\frac{9}{2},\frac{13}{2}) \) or \( (4.5,6.5) \).

Answer:

The solution of the system of equations is \( x = \frac{9}{2},y=\frac{13}{2} \) (or \( (4.5,6.5) \))