To find the point of intersection of the two lines given by the equations \( y = 4x - 3 \) and \( y = -2x + 9 \), we can set the right-hand sides of the equations equal to each other since they are both equal to \( y \).
So we set:
\[ 4x - 3 = -2x + 9 \]
Now, we can solve for \( x \):
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Add \( 2x \) to both sides: \[ 4x + 2x - 3 = 9 \] \[ 6x - 3 = 9 \]
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Add \( 3 \) to both sides: \[ 6x = 12 \]
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Divide by \( 6 \): \[ x = 2 \]
Now that we have the value of \( x \), we can substitute it back into either of the original equations to find \( y \). We'll use the first equation \( y = 4x - 3 \):
\[ y = 4(2) - 3 \] \[ y = 8 - 3 \] \[ y = 5 \]
Thus, the point of intersection of the two lines is:
\[ \boxed{(2, 5)} \]
You can also verify by substituting \( x = 2 \) into the second equation \( y = -2x + 9 \):
\[ y = -2(2) + 9 \] \[ y = -4 + 9 \] \[ y = 5 \]
Hence, the intersection point \( (2, 5) \) is confirmed.