Question

how many solutions
y = 3x + 6
y = -13x - 4

Answer 1

To find the number of solutions to the system of equations:

1. \( y = 3x + 6 \)
2. \( y = -13x - 4 \)

we can set the two equations equal to each other since they both represent \( y \).

So we set:

\[
3x + 6 = -13x - 4
\]

Now, let's solve for \( x \):

1. Add \( 13x \) to both sides:
\[
3x + 13x + 6 = -4
\]
This simplifies to:
\[
16x + 6 = -4
\]

2. Next, subtract 6 from both sides:
\[
16x = -4 - 6
\]
Which simplifies to:
\[
16x = -10
\]

3. Finally, divide both sides by 16:
\[
x = -\frac{10}{16} = -\frac{5}{8}
\]

Now, we can substitute \( x = -\frac{5}{8} \) back into either of the original equations to find \( y \). Let's substitute it into the first equation:

\[
y = 3\left(-\frac{5}{8}\right) + 6
\]

Calculating this:

\[
y = -\frac{15}{8} + 6
\]
Convert 6 into eighths:
\[
6 = \frac{48}{8}
\]
So,
\[
y = -\frac{15}{8} + \frac{48}{8} = \frac{33}{8}
\]

Thus, the point of intersection (the solution) is:
\[
\left(-\frac{5}{8}, \frac{33}{8}\right)
\]

Since we have found one unique point of intersection, the system of equations has exactly **one solution**.

Therefore, the answer is: **1 solution**.