To solve the system of equations given by:
- \( y = |x| - 1 \)
- \( y = x + 3 \)
we need to find the points where these two graphs intersect.
Step 1: Set the equations equal to each other:
\[ |x| - 1 = x + 3 \]
Step 2: Solve the absolute value equation:
Case 1: \( x \geq 0 \)
In this case, \( |x| = x \). The equation becomes:
\[ x - 1 = x + 3 \]
Subtracting \( x \) from both sides:
\[ -1 = 3 \]
This is not true, so there are no solutions in this case.
Case 2: \( x < 0 \)
In this case, \( |x| = -x \). The equation becomes:
\[ -x - 1 = x + 3 \]
Adding \( x \) to both sides:
\[ -1 = 2x + 3 \]
Subtracting 3 from both sides:
\[ -4 = 2x \]
Dividing by 2:
\[ x = -2 \]
Step 3: Find \( y \) using either original equation:
Using \( y = x + 3 \):
\[ y = -2 + 3 = 1 \]
Conclusion:
The solution to the system of equations is:
\[ \boxed{(-2, 1)} \]
Thus, the intersection point, or the solution to the system, is at \((-2, 1)\).
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