To find the solution of the equations \( \frac{7}{2}x - \frac{9}{2} = 2y \) and \( 4y = 7x - 9 \), we first need to examine both equations.
Let's manipulate the first equation:
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Start with: \[ \frac{7}{2}x - \frac{9}{2} = 2y \]
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Rearrange it to isolate \( y \): \[ 2y = \frac{7}{2}x - \frac{9}{2} \] Divide both sides by 2: \[ y = \frac{7}{4}x - \frac{9}{4} \]
Now, let's manipulate the second equation:
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Start with: \[ 4y = 7x - 9 \]
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Rearrange it to isolate \( y \): \[ y = \frac{7}{4}x - \frac{9}{4} \]
Now we have both equations in slope-intercept form:
- From the first equation: \( y = \frac{7}{4}x - \frac{9}{4} \)
- From the second equation: \( y = \frac{7}{4}x - \frac{9}{4} \)
Since both equations are the same, they represent the same line on a graph.
To find the solutions of the equation system, we can set the two expressions for \( y \) equal to each other, but since they are identical, we can find their intersections by plugging in any value of \( x \) to find the corresponding value of \( y \).
To find a particular solution, we can set \( x = 0 \):
- Substitute \( x = 0 \): \[ y = \frac{7}{4}(0) - \frac{9}{4} = -\frac{9}{4} \]
Thus, a specific point on the line is \( (0, -\frac{9}{4}) \).
Alternatively, let's find another point by setting \( y = 0 \):
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Set \( y = 0 \): \[ 0 = \frac{7}{4}x - \frac{9}{4} \]
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Rearranging this gives: \[ \frac{7}{4}x = \frac{9}{4} \] Multiplying by \( \frac{4}{7} \): \[ x = \frac{9}{7} \]
Now, we found two solutions:
- \( (0, -\frac{9}{4}) \)
- \( (\frac{9}{7}, 0) \)
The lines intersect infinitely at all points on the line \( y = \frac{7}{4}x - \frac{9}{4} \).
Thus, the solution set of the system of equations is represented by the line, or \( {(x, \frac{7}{4}x - \frac{9}{4}) : x \in \mathbb{R}} \).