To determine the solution of the equations \( (72)x - (92) = 2y \) and \( 4y = 7x - 9 \), we can start by rearranging both equations into slope-intercept form (y = mx + b) and see if they are equivalent.
Starting with the first equation:
- Rearrange \( (72)x - (92) = 2y \): \[ 2y = 72x - 92 \] \[ y = 36x - 46 \]
Next, for the second equation:
- Rearrange \( 4y = 7x - 9 \): \[ y = \frac{7}{4}x - \frac{9}{4} \]
Now we check if both equations represent the same line. We observe their slopes (36 and 1.75) and y-intercepts (-46 and -2.25). Since these values are not equal, the lines represented by these equations are not the same.
Since they intersect at a specific point, we can find that point by setting the equations equal to each other.
Set the y-values equal to each other (as we have both in y = mx + b form): \[ 36x - 46 = \frac{7}{4}x - \frac{9}{4} \]
To eliminate the fraction, multiply through by 4: \[ 4(36x - 46) = 7x - 9 \] \[ 144x - 184 = 7x - 9 \]
Now, combine like terms: \[ 144x - 7x = -9 + 184 \] \[ 137x = 175 \] \[ x = \frac{175}{137} \approx 1.28 \]
Now substitute \( x \) back into either equation to find \( y \): Using \( y = 36x - 46 \): \[ y = 36\left(\frac{175}{137}\right) - 46 \] Calculating it yields \( y \approx 0 \).
Thus, the intersection point is roughly \( (1.286, 0) \).
Therefore, since both equations yield the same line when reduced, the final answer is that they intersect at one specific point:
(1.286, 0)
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