Question

e the following system of equations graphically on the set of axes b
( y = x + 3 )
( 3x + y = -5 )
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Analyze the first equation \( y = x + 3 \)

This is in slope - intercept form \( y=mx + b \), where the slope \( m = 1 \) and the y - intercept \( b=3 \). To graph this line, we can start by plotting the y - intercept at the point \( (0,3) \). Then, using the slope (rise over run), since the slope is 1, we can move 1 unit up and 1 unit to the right from \( (0,3) \) to get the point \( (1,4) \), or 1 unit down and 1 unit to the left to get the point \( (- 1,2) \).

Step2: Analyze the second equation \( 3x + y=-5 \)

We can rewrite this equation in slope - intercept form (\( y=mx + b \)) by solving for \( y \). Subtract \( 3x \) from both sides: \( y=-3x - 5 \). Here, the slope \( m=-3 \) and the y - intercept \( b = - 5 \). To graph this line, we start by plotting the y - intercept at the point \( (0,-5) \). Then, using the slope, since the slope is - 3 (which can be thought of as \( \frac{-3}{1} \) or \( \frac{3}{-1} \)), we can move 3 units down and 1 unit to the right from \( (0,-5) \) to get the point \( (1,-8) \), or 3 units up and 1 unit to the left to get the point \( (-1,-2) \).

Step3: Find the intersection point (solution of the system)

We can also solve the system algebraically to find the intersection point. Substitute \( y=x + 3 \) into \( 3x + y=-5 \):
\[

$$\begin{align*} 3x+(x + 3)&=-5\\ 3x+x+3&=-5\\ 4x&=-5 - 3\\ 4x&=-8\\ x&=-2 \end{align*}$$

\]
Then substitute \( x = - 2 \) into \( y=x + 3 \), we get \( y=-2 + 3=1 \). So the two lines intersect at the point \( (-2,1) \). When graphing, the first line \( y=x + 3 \) passes through \( (0,3),(1,4),(-1,2) \) etc., and the second line \( y=-3x - 5 \) passes through \( (0,-5),(1,-8),(-1,-2) \) etc. The point of intersection is \( (-2,1) \).

Answer:

To graph the lines:

• For \( y=x + 3 \): Plot \( (0,3) \) and use the slope \( 1 \) to find other points.
• For \( 3x + y=-5 \) (or \( y=-3x - 5 \)): Plot \( (0,-5) \) and use the slope \( - 3 \) to find other points. The intersection point (solution) is \( (-2,1) \). If we were to plot the lines on the given axes, we would mark these points and draw the lines through them. The final answer for the solution of the system (the point where the two lines intersect) is \( x=-2,y = 1 \) or the ordered pair \( (-2,1) \).