To find the number of solutions to the system of equations:
- \( y = 3x + 6 \)
- \( y = -13x - 4 \)
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 \( x \):
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Add \( 13x \) to both sides: \[ 3x + 13x + 6 = -4 \] This simplifies to: \[ 16x + 6 = -4 \]
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Next, subtract 6 from both sides: \[ 16x = -4 - 6 \] Which simplifies to: \[ 16x = -10 \]
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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.