Question

solve the following system of equations graphically on the set of axes below.
$y = x + 3$
$y = -dfrac{1}{5}x - 3$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Find two points for \( y = x + 3 \)

When \( x = 0 \), \( y = 0 + 3 = 3 \), so we have the point \( (0, 3) \).
When \( x = -3 \), \( y = -3 + 3 = 0 \), so we have the point \( (-3, 0) \).

Step2: Find two points for \( y = -\frac{1}{5}x - 3 \)

When \( x = 0 \), \( y = -\frac{1}{5}(0) - 3 = -3 \), so we have the point \( (0, -3) \).
When \( x = 5 \), \( y = -\frac{1}{5}(5) - 3 = -1 - 3 = -4 \)? Wait, no, let's recalculate. Wait, when \( x = 5 \), \( y = -\frac{1}{5}(5) - 3 = -1 - 3 = -4 \)? Wait, no, maybe better to use \( x = -5 \): \( y = -\frac{1}{5}(-5) - 3 = 1 - 3 = -2 \)? Wait, no, let's do it properly. Let's find \( x \) when \( y = 0 \): \( 0 = -\frac{1}{5}x - 3 \Rightarrow \frac{1}{5}x = -3 \Rightarrow x = -15 \). But that's too far. Alternatively, use \( x = 5 \): \( y = -\frac{1}{5}(5) - 3 = -1 - 3 = -4 \), so point \( (5, -4) \). Or \( x = -5 \): \( y = -\frac{1}{5}(-5) - 3 = 1 - 3 = -2 \), so point \( (-5, -2) \).

But maybe instead of plotting, we can solve the system algebraically to find the intersection, then plot.

Step3: Solve the system algebraically (to find intersection)

Set \( x + 3 = -\frac{1}{5}x - 3 \)
Add \( \frac{1}{5}x \) to both sides: \( x + \frac{1}{5}x + 3 = -3 \)
Combine like terms: \( \frac{6}{5}x + 3 = -3 \)
Subtract 3: \( \frac{6}{5}x = -6 \)
Multiply both sides by \( \frac{5}{6} \): \( x = -6 \times \frac{5}{6} = -5 \)

Then \( y = x + 3 = -5 + 3 = -2 \)

So the intersection point is \( (-5, -2) \).

Now, to plot the lines:

• For \( y = x + 3 \), plot \( (0, 3) \) and \( (-3, 0) \), then draw the line through them.
• For \( y = -\frac{1}{5}x - 3 \), plot \( (0, -3) \) and \( (-5, -2) \) (wait, when \( x = -5 \), \( y = -\frac{1}{5}(-5) - 3 = 1 - 3 = -2 \), yes, that's correct). So plot \( (0, -3) \) and \( (-5, -2) \), then draw the line through them.

The intersection point is \( (-5, -2) \), which is the solution.

Answer:

The solution to the system is \( x = -5 \), \( y = -2 \), so the point \( (-5, -2) \). When plotting, the two lines intersect at \( (-5, -2) \).