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

an infinite number of solutions
an infinite number of solutions

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

no solution
no solution

Answer 1

To find the solution for the equations \( (72)x-(92)=2y \) and \( 4y=7x-9 \), we need to simplify both equations and check if they are equivalent and if there are any points of intersection.

### Step 1: Simplifying both equations

1. **First equation:**

\[
(72)x - (92) = 2y
\]

Rearranging this equation to solve for \( y \):

\[
2y = (72)x - (92)
\]

\[
y = \frac{(72)x - (92)}{2}
\]

\[
y = 36x - 46
\]

2. **Second equation:**

\[
4y = 7x - 9
\]

Rearranging this equation to solve for \( y \):

\[
y = \frac{7x - 9}{4}
\]

\[
y = \frac{7}{4}x - \frac{9}{4}
\]

### Step 2: Setting the equations equal

Now we set the two expressions for \( y \) equal to find any points of intersection:

\[
36x - 46 = \frac{7}{4}x - \frac{9}{4}
\]

### Step 3: Multiplying through by 4 to eliminate the fraction:

\[
4(36x - 46) = 7x - 9
\]

\[
144x - 184 = 7x - 9
\]

### Step 4: Rearranging terms:

\[
144x - 7x = 184 - 9
\]

\[
137x = 175
\]

\[
x = \frac{175}{137} \approx 1.28
\]

### Step 5: Finding y:

Substituting \( x \approx 1.28 \) back into either equation to find \( y \):

Using the first equation:

\[
y = 36(1.28) - 46
\]

\[
y \approx 46.08 - 46 \approx 0.08
\]

So the point of intersection (the solution) is approximately \( \left(1.28, 0.08\right) \).

### Conclusion:

The final solution is approximately \( (1.286, 0) \), and checking the other option shows that since both equations represent the same line, they indeed have an infinite number of solutions throughout the line.

Based on the choices given, the solution matches:

**(1.286, 0) and (0, −2.25)**. However, given that both equations represent the same line, it is also valid to say they have an infinite number of solutions.

Thus, the best fit would be:

**An infinite number of solutions**