Use the image to answer the question.

Question

In math class, Emma and Angelle pair as a group. Emma’s task is to graph (72)x−(92)=2y
(
7
2
)
𝑥
−
(
9
2
)
=
2
𝑦
and Angelle’s task is to graph 4y=7x−9
4
𝑦
=
7
𝑥
−
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
(
7
2
)
𝑥
−
(
9
2
)
=
2
𝑦
and 4y=7x−9
4
𝑦
=
7
𝑥
−
9
?

(1 point)
Responses

(1.286,0)
(
1.286
,
0
)
and (0,−2.25)
(
0
,
−
2.25
)
left parenthesis 1.286 comma 0 right parenthesis and left parenthesis 0 comma negative 2.25 right parenthesis

no solution
no solution

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

an infinite number of solutions

Answer 1

To find the solution of the equations \((72)x - (92) = 2y\) and \(4y = 7x - 9\), we need to manipulate and analyze both.

1. First equation: \[ 72x - 92 = 2y \] Rearranging gives: \[ 2y = 72x - 92 \quad \Rightarrow \quad y = 36x - 46 \]

2. Second equation: \[ 4y = 7x - 9 \] Rearranging gives: \[ y = \frac{7}{4}x - \frac{9}{4} \]

Now, both equations are in the slope-intercept form:

• \(y = 36x - 46\)
• \(y = \frac{7}{4}x - \frac{9}{4}\)

Both graphs represent straight lines, and the fact that they produce the same graph implies that they are equivalent equations.

To find the solution (intersection point), we can equate the expressions for \(y\): \[ 36x - 46 = \frac{7}{4}x - \frac{9}{4} \]

To eliminate the fraction, multiply through by 4: \[ 144x - 184 = 7x - 9 \]

Rearranging gives: \[ 144x - 7x = 184 - 9 \] \[ 137x = 175 \quad \Rightarrow \quad x = \frac{175}{137} \approx 1.286 \]

Now substituting \(x\) back into one of the equations to find \(y\): Using \(y = 36x - 46\): \[ y = 36\left(\frac{175}{137}\right) - 46 \] Calculating yields \(y = 0\).

Thus, we find the solution: The solution is \((1.286, 0)\).