solve the following system of equations graphically on the set of axes below. \n$y = -dfrac{1}{2}x + 7$\n$y…

Explanation:

Step1: Analyze the first equation

The equation ( y = -\frac{1}{2}x + 7 ) is in slope - intercept form (( y=mx + b )), where the slope ( m=-\frac{1}{2} ) and the y - intercept ( b = 7 ). To graph this line, we start by plotting the y - intercept at the point ( (0,7) ). Then, using the slope, from the point ( (0,7) ), we move down 1 unit and to the right 2 units (since slope (=\frac{\text{rise}}{\text{run}}=-\frac{1}{2}=\frac{- 1}{2})) to get another point, say ( (2,6) ), and draw the line through these points.

Step2: Analyze the second equation

The equation ( y=\frac{3}{2}x - 5 ) is also in slope - intercept form. The slope ( m = \frac{3}{2} ) and the y - intercept ( b=-5 ). We start by plotting the y - intercept at the point ( (0, - 5) ). Then, using the slope, from the point ( (0,-5) ), we move up 3 units and to the right 2 units (since slope (=\frac{\text{rise}}{\text{run}}=\frac{3}{2})) to get another point, say ( (2,-2) ), and draw the line through these points.

Step3: Find the intersection point

To solve the system graphically, we find the point where the two lines intersect. We can also solve the system algebraically to verify. Set the two equations equal to each other: [ -\frac{1}{2}x + 7=\frac{3}{2}x-5 ] Add ( \frac{1}{2}x ) to both sides: [ 7=\frac{3}{2}x+\frac{1}{2}x - 5 ] [ 7 = 2x-5 ] Add 5 to both sides: [ 12 = 2x ] Divide both sides by 2: [ x = 6 ] Substitute ( x = 6 ) into the first equation ( y=-\frac{1}{2}(6)+7=-3 + 7=4 )

So the intersection point (the solution of the system) is ( (6,4) ). When graphing, the two lines will intersect at the point ( (6,4) ).

Answer:

The solution of the system of equations is ( x = 6,y = 4 ) or the point ( (6,4) )