To determine a linear equation that results in a system with exactly one solution when graphed alongside the given line, we need to ensure that the new line intersects the existing line at a single point. This requires that the new line has a different slope than that of the original line.
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Identify the slope of the original line: We can find the slope (m) of the line using two of the given points. The slope formula is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Let's use the points (0, -2) and (3, -1): \[ m = \frac{-1 - (-2)}{3 - 0} = \frac{1}{3} \]
Therefore, the slope of the original line is \(\frac{1}{3}\).
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Find an equation with a different slope: To ensure that we get exactly one solution, we need to choose an equation with a different slope. Without specific equations provided, choose any line that has a slope not equal to \(\frac{1}{3}\).
For example:
- If you choose a line with a slope of 1, the equation could be \(y = x + b\) for some value of b that ensures it intersects the original line.
- If you choose a line with a slope of -1, the equation could be \(y = -x + c\).
- Verify the choice: Make sure to plot the new line on the graph. If it intersects at exactly one point, then that is the desired equation.
Conclusion:
Without the specific options A, B, C, and D, ensure that the new equation has a slope different from \(\frac{1}{3}\). For the situation described, lines with slopes such as 1, -1, or any slope other than \(\frac{1}{3}\) will yield a system with exactly one solution. If certain options were provided, find one with a slope that differs from \(\frac{1}{3}\).