Question
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3. \( 2 x-3 y=4 \) and \( 4 x-3 y=20 \)

Ask by Harmon Lyons. in South Africa
Mar 09,2025

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Tutor-Verified Answer

Answer

\( x = 8 \) and \( y = 4 \).

Solution

1. Start with the given equations: \[ 2x - 3y = 4 \] \[ 4x - 3y = 20 \] 2. Subtract the first equation from the second to eliminate \(y\): \[ (4x - 3y) - (2x - 3y) = 20 - 4 \] \[ 4x - 3y - 2x + 3y = 16 \] \[ 2x = 16 \] 3. Solve for \(x\): \[ x = \frac{16}{2} = 8 \] 4. Substitute \(x = 8\) back into the first equation: \[ 2(8) - 3y = 4 \] \[ 16 - 3y = 4 \] 5. Solve for \(y\): \[ -3y = 4 - 16 \] \[ -3y = -12 \] \[ y = \frac{-12}{-3} = 4 \] The solution to the system is \(x = 8\) and \(y = 4\).

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Beyond the Answer

Both of these equations represent straight lines on a Cartesian plane, and they can be solved simultaneously to find their intersection point. To solve these equations, you can use substitution or elimination. Notably, the second equation can be modified to reveal that the two equations are actually aligned since rearranging it yields a similar form to the first equation, hinting at potential parallel lines or coincidence. When solving systems of equations, one common mistake is to overlook the possibility of lines being parallel (having no solution) or coincident (infinitely many solutions). In this case, it's worthwhile to simplify the equations properly and check for unique or special solutions before proceeding to graphing or applying more complex methods!

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