To estimate the solution to the system of equations given by \( y = 2x + 3 \) and \( y = 12x + 6 \), we need to find the intersection point of these two lines.
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First, we can set the two equations equal to each other: \[ 2x + 3 = 12x + 6 \]
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Rearranging the equation to isolate terms involving \( x \): \[ 3 - 6 = 12x - 2x \] \[ -3 = 10x \] \[ x = -\frac{3}{10} \]
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Now, substitute \( x \) back into one of the original equations to find \( y \). We'll use \( y = 2x + 3 \): \[ y = 2\left(-\frac{3}{10}\right) + 3 \] \[ y = -\frac{6}{10} + 3 \] \[ y = -\frac{6}{10} + \frac{30}{10} \] \[ y = \frac{24}{10} = \frac{12}{5} \]
Thus, the solution (intersection point) of the system of equations is approximately \( (-0.3, 2.4) \).
Now looking at the provided responses, none matches the calculated solution precisely. However, based on the options provided:
- \( (2, 7) \)
- \( (2, -7) \)
- \( (7, 2) \)
- \( (-2, 7) \)
Let's estimate those values based on the original equations.
If we check \( (2, 7) \):
- For \( y = 2(2) + 3 = 7 \) (correct)
- For \( y = 12(2) + 6 = 30 \) (not correct)
If we check \( (2, -7) \):
- For \( y = 2(2) + 3 = 7 \) (correct)
- For \( y = 12(2) + 6 = 30 \) (not correct)
If we check \( (7, 2) \):
- For \( y = 2(7) + 3 = 17 \) (not correct)
- For \( y = 12(7) + 6 = 90 \) (not correct)
If we check \( (-2, 7) \):
- For \( y = 2(-2) + 3 = -4 + 3 = -1 \) (not correct)
- For \( y = 12(-2) + 6 = -24 + 6 = -18 \) (not correct)
Given that none of the points satisfy both equations, the closest point that fits the higher value for one of the equations can be estimated and would be likely \( (2, 7) \), answering based on the approximation.
Therefore, the best fitting option would be: (2, 7).