To find the solution to the system of equations given by \( y = 2x - 1 \) and \( y - 3 = 4x \), we can start by rewriting the second equation in slope-intercept form.
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Rewrite the second equation:
\[ y - 3 = 4x \] Adding 3 to both sides: \[ y = 4x + 3 \]
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Set the two equations equal to each other:
Now we have: \[ y = 2x - 1 \] \[ y = 4x + 3 \]
We can set these equations equal to each other: \[ 2x - 1 = 4x + 3 \]
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Solve for x:
Rearranging the equation: \[ 2x - 4x = 3 + 1 \] \[ -2x = 4 \] Dividing both sides by -2: \[ x = -2 \]
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Substitute x back to find y:
Now we can substitute \( x = -2 \) back into either equation to find \( y \). We'll use \( y = 2x - 1 \): \[ y = 2(-2) - 1 \] \[ y = -4 - 1 \] \[ y = -5 \]
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Result:
The ordered pair that is the solution to the system of equations is: \[ (-2, -5) \]
Thus, the ordered pair \((-2, -5)\) is the solution to the system of equations because it is the point at which both lines intersect, so both lines share this point.
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