solve the following system of equations graphically on the set of axes below.\n$y = -2x - 3$\n$y =…

Explanation:

Step1: Graph ( y = -2x - 3 )

The equation is in slope - intercept form ( y=mx + b ), where ( m=-2 ) (slope) and ( b = - 3 ) (y - intercept). Plot the y - intercept at ( (0,-3) ). Then, use the slope: from ( (0,-3) ), move down 2 units and right 1 unit (or up 2 units and left 1 unit) to find another point. For example, when ( x = 0 ), ( y=-3 ); when ( x=-1 ), ( y=-2\times(-1)-3=2 - 3=-1 ), so the point ( (-1,-1) ) is also on the line.

Step2: Graph ( y=\frac{1}{3}x + 4 )

This is also in slope - intercept form with ( m=\frac{1}{3} ) (slope) and ( b = 4 ) (y - intercept). Plot the y - intercept at ( (0,4) ). Then, use the slope: from ( (0,4) ), move up 1 unit and right 3 units (or down 1 unit and left 3 units) to find another point. For example, when ( x = 0 ), ( y = 4 ); when ( x=3 ), ( y=\frac{1}{3}\times3 + 4=1 + 4 = 5 ), so the point ( (3,5) ) is on the line.

Step3: Find the intersection

The solution to the system of equations is the point where the two lines intersect. To find the intersection algebraically (to confirm the graphical solution), set the two equations equal to each other: [ \begin{align*} -2x-3&=\frac{1}{3}x + 4\ -2x-\frac{1}{3}x&=4 + 3\ -\frac{6x+x}{3}&=7\ -\frac{7x}{3}&=7\ x&=7\times\left(-\frac{3}{7}\right)\ x&=- 3 \end{align*} ] Substitute ( x=-3 ) into ( y=-2x - 3 ): ( y=-2\times(-3)-3=6 - 3 = 3 ). So the intersection point is ( (-3,3) ). When graphing, the two lines will cross at ( (-3,3) ).

Answer:

The solution to the system of equations is ( x=-3,y = 3 ) (or the point ( (-3,3) )).