Bot GPT 4-o mini To determine the solution of the two equations \( (72)x - (92) = 2y \) and \( 4y = 7x - 9 \), first we will rearrange both equations into slope-intercept form (i.e., \( y = mx + b \)).
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For the first equation: \[ (72)x - (92) = 2y \] To isolate \( y \), first add 92 to both sides: \[ 72x - 92 = 2y \implies 2y = 72x - 92 \] Now divide every term by 2: \[ y = 36x - 46 \]
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For the second equation: \[ 4y = 7x - 9 \] Now, divide every term by 4: \[ y = \frac{7}{4}x - \frac{9}{4} \]
Now we have two equations:
- \( y = 36x - 46 \)
- \( y = \frac{7}{4}x - \frac{9}{4} \)
Since Emma and Angelle got exactly the same graph, it indicates that the two equations are equivalent.
To find the solution, we can set the two equations equal to each other:
\[ 36x - 46 = \frac{7}{4}x - \frac{9}{4} \]
To eliminate the fraction, multiply the entire equation by 4: \[ 4(36x) - 4(46) = 7x - 9 \] This simplifies to: \[ 144x - 184 = 7x - 9 \] Now, isolate the variable \( x \): \[ 144x - 7x = 184 - 9 \] \[ 137x = 175 \] \[ x = \frac{175}{137} \approx 1.286 \]
Next, substitute \( x \) back to find \( y \): Using the equation \( y = 36x - 46 \): \[ y = 36\left(\frac{175}{137}\right) - 46 \] Calculating: \[ y \approx 36 \cdot 1.286 - 46 \approx 46.296 - 46 = 0.296 \text{ (not exact since giving decimals)} \] Now we check back:
Checking in \( 4y = 7x - 9 \): Substituting \( x \approx 1.286 \) gives: \[ y = \frac{7}{4}(1.286) - \frac{9}{4} \approx 2.245 - 2.25 \approx 0 \]
Thus, the solution point is \( (1.286, 0) \).
Since both graphs are the same, the solution is either that they intersect at one point or an infinite number of solutions exist if they describe the same line (verified both are equivalent).
Since both equations yield only the point of intersection:
The answer is: (1.286, 0).
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