To determine the number of solutions to the given system of equations, we can analyze it algebraically and graphically.
Equations:
- \( y = x + 3 \) (Equation 1)
- \( 3x + y = -1 \) (Equation 2)
Algebraic Method:
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Substitute Equation 1 into Equation 2:
We can substitute \( y \) from Equation 1 into Equation 2: \[ 3x + (x + 3) = -1 \] Simplifying this: \[ 4x + 3 = -1 \] \[ 4x = -1 - 3 \] \[ 4x = -4 \] \[ x = -1 \]
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Substitute \( x = -1 \) back into Equation 1 to find \( y \): \[ y = -1 + 3 = 2 \] So, the solution to the system is \( (x, y) = (-1, 2) \).
Since we found one unique solution, the system of equations has exactly one solution.
Graphical Method:
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Graphing the equations:
- For \( y = x + 3 \), the line has a slope of 1 and a y-intercept at (0, 3). It goes up one unit for each unit it goes to the right.
- For \( 3x + y = -1 \), we can rearrange it to slope-intercept form: \[ y = -3x - 1 \] The slope is -3, and the y-intercept is (0, -1). This line goes down three units for each unit it goes to the right.
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Analyzing the graphs:
- When you graph both lines, you'll see that they intersect at the point (-1, 2). Since they intersect at a single point, this signifies that there is exactly one solution.
Conclusion:
- From the equations: We found one unique solution by substituting and solving.
- From the graph: The two lines intersect at one point, indicating one solution.
Thus, the system of equations has exactly one solution at the point \((-1, 2)\).