Question

solve the following system of equations graphically on the set of axes below.
$y = -3x - 7$
$y = x - 3$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Graph \( y = -3x - 7 \)

To graph a linear equation in slope - intercept form \( y=mx + b \) (where \( m \) is the slope and \( b \) is the y - intercept), for the equation \( y=-3x - 7 \), the y - intercept \( b=-7 \), so we start by plotting the point \( (0,-7) \) on the y - axis. The slope \( m=-3=\frac{-3}{1} \), which means from the point \( (0, - 7) \), we move down 3 units (because the numerator of the slope is - 3) and then 1 unit to the right (because the denominator of the slope is 1) to get the next point \( (1,-10) \), or we can move up 3 units and 1 unit to the left to get the point \( (- 1,-4) \). We can use these two points to draw the line.

Step2: Graph \( y=x - 3 \)

For the equation \( y=x - 3 \), the slope - intercept form is \( y = 1x-3 \), so the y - intercept \( b=-3 \), and we plot the point \( (0,-3) \) on the y - axis. The slope \( m = 1=\frac{1}{1} \), so from the point \( (0,-3) \), we move up 1 unit and 1 unit to the right to get the point \( (1,-2) \), or down 1 unit and 1 unit to the left to get the point \( (-1,-4) \). We use these two points to draw the line.

Step3: Find the intersection point

The solution to the system of linear equations is the point where the two lines intersect. By looking at the two lines we have graphed, we can see that they intersect at the point \( (-1,-4) \). We can also verify this algebraically by setting the two equations equal to each other:

$$
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$$

Substitute \( x = - 1 \) into \( y=x - 3 \), we get \( y=-1-3=-4 \).

Answer:

The solution to the system of equations is \( x=-1,y = - 4 \), or the point \( (-1,-4) \)