In math class, Emma and Angelle pair as a group. Emma’s task is to graph (72)x−(92)=2y and Angelle’s task is to graph 4y=7x−9 . After they complete their graphs, they compare their work. They both get exactly the same graph. What is the solution of (72)x−(92)=2y and 4y=7x−9 ?

Question

(1 point)
Responses

no solution
no solution

(1.286,0)
left parenthesis 1.286 comma 0 right parenthesis

an infinite number of solutions
an infinite number of solutions

(1.286,0) and (0,−2.25)

Answer 1

To find the solution of the equations \( (72)x - (92) = 2y \) and \( 4y = 7x - 9 \), we will first simplify both equations to check if they are equivalent.

1. Start with the first equation: \[ (72)x - (92) = 2y \] Rearranging this gives: \[ 2y = 72x - 92 \] Now, divide everything by 2: \[ y = 36x - 46 \]

2. Next, simplify the second equation: \[ 4y = 7x - 9 \] Now, divide everything by 4: \[ y = \frac{7}{4}x - \frac{9}{4} \]

Now we have two equations:

1. \( y = 36x - 46 \)
2. \( y = \frac{7}{4}x - \frac{9}{4} \)

Now we check if these functions represent the same line by equating them: \[ 36x - 46 = \frac{7}{4}x - \frac{9}{4} \]

To solve for \(x\), first eliminate the fraction by multiplying the entire equation by 4: \[ 144x - 184 = 7x - 9 \]

Now, rearranging gives: \[ 144x - 7x = 184 - 9 \] \[ 137x = 175 \] \[ x = \frac{175}{137} \approx 1.286 \]

Now substitute \(x\) back into either equation to find \(y\). Using the first equation: \[ y = 36\left(\frac{175}{137}\right) - 46 \] Calculating that gives: \[ y = \frac{6300}{137} - 46 = \frac{6300 - 6302}{137} = \frac{-2}{137} \approx 0 \]

Therefore, the point of intersection (solution) is \( \left(\frac{175}{137}, 0\right) \).

Since both equations are different forms of the same line and they've grappled with definite slope and intercepts yielding a single intersection, we conclude:

The correct response is: (1.286, 0).