To determine the number of solutions to the system of linear equations represented by the two lines, we need to analyze their slopes and their intercepts based on the given points.
First Line:
- It passes through the points (0, 2) and (1, 0).
- The slope (m) of the line can be calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 2}{1 - 0} = \frac{-2}{1} = -2 \]
- The y-intercept \(b\) is at (0, 2), so the equation of the first line is: \[ y = -2x + 2 \]
Second Line:
- It passes through the points (0, -1) and (1, -2).
- The slope of the line can be calculated as: \[ m = \frac{-2 - (-1)}{1 - 0} = \frac{-2 + 1}{1} = \frac{-1}{1} = -1 \]
- The y-intercept \(b\) is at (0, -1), so the equation of the second line is: \[ y = -x - 1 \]
Finding the Point of Intersection: To find the number of solutions to the system, we can find the point where the two lines intersect by setting their equations equal to each other: \[ -2x + 2 = -x - 1 \] Now, solve for \(x\):
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Rearranging gives: \[ -2x + x = -1 - 2 \] \[ -x = -3 \] \[ x = 3 \]
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Substitute \(x = 3\) back into one of the equations to find \(y\): Using \(y = -x - 1\): \[ y = -3 - 1 = -4 \]
So the lines intersect at the point \( (3, -4) \).
Conclusions on Solutions:
- The lines intersect at a single point, which means there is exactly one solution to the system of equations.
Therefore, the final answer is: One solution at (3, -4).