To estimate the solution to the system of equations given by \(y = 2x + 3\) and \(y = 12x + 6\), you need to find the point where the two lines intersect.
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Set the two equations equal to each other to find the \(x\)-coordinate of the intersection: \[ 2x + 3 = 12x + 6 \]
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Rearranging the equation: \[ 3 - 6 = 12x - 2x \] \[ -3 = 10x \] \[ x = -\frac{3}{10} \]
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Substitute \(x = -\frac{3}{10}\) back into either equation to find the corresponding \(y\)-coordinate. Using the first equation: \[ y = 2\left(-\frac{3}{10}\right) + 3 \] \[ y = -\frac{6}{10} + 3 = -\frac{6}{10} + \frac{30}{10} = \frac{24}{10} = \frac{12}{5} = 2.4 \]
Therefore, the intersection point is approximately \((-0.3, 2.4)\).
Since the options given in the responses are discrete points, we need to check which one might be the closest estimate based on the position of the lines:
The options were:
- (2, 7)
- (2, -7)
- (7, 2)
- (-2, 7)
The calculated intersection does not match any of the points exactly, but the closest point based on estimates should be checked graphically.
Given the steepness of the second line, it suggests the intersection is very close to around where \( (-2, 7) \) is, as it appears both lines would cross around there (but this is contextually an estimate based on the slopes).
Thus, if choosing based on estimates from a graph, I would suggest: (−2, 7).
Make sure to visualize or calculate depending on further data or a graph representation.
1 answer