To estimate the solution to the system of the linear equations \(y = 2x + 3\) and \(y = \frac{1}{2}x + 6\), we can find the point where the two lines intersect.
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Set the equations equal to each other since both are equal to \(y\): \[ 2x + 3 = \frac{1}{2}x + 6 \]
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Clear the fraction by multiplying all terms by 2 to eliminate the fraction: \[ 4x + 6 = x + 12 \]
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Rearrange the equation to isolate \(x\): \[ 4x - x = 12 - 6 \] \[ 3x = 6 \] \[ x = 2 \]
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Substitute \(x = 2\) back into one of the original equations to find \(y\). We can use either equation; let’s use \(y = 2x + 3\): \[ y = 2(2) + 3 = 4 + 3 = 7 \]
Thus, the solution to the system of equations is \(x = 2\) and \(y = 7\), which gives the point of intersection as \((2, 7)\).
In conclusion, the estimated solution is \((2, 7)\).