Question

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 ? (1 point) Responses (1.286,0) left parenthesis 1.286 comma 0 right parenthesis an infinite number of solutions an infinite number of solutions no solution no solution (1.286,0) and (0,−2.25)

Answer 1

To determine the solution to the equations \( (72)x - (92) = 2y \) and \( 4y = 7x - 9 \), we first need to manipulate both equations to understand their graphs.

### Step 1: Rearranging the first equation
Starting with the first equation:
\[
(72)x - (92) = 2y
\]
We can rearrange it to solve for \( y \):
\[
2y = (72)x - (92)
\]
Dividing everything by 2:
\[
y = \frac{72}{2}x - \frac{92}{2}
\]
\[
y = 36x - 46
\]

### Step 2: Rearranging the second equation
Now, we take the second equation:
\[
4y = 7x - 9
\]
Rearranging it to solve for \( y \):
\[
y = \frac{7}{4}x - \frac{9}{4}
\]

### Step 3: Finding intersection points or solutions
To check if these two lines are the same, we can see if their slopes and y-intercepts match.

For the first line:
- Slope = 36
- Y-intercept = -46

For the second line:
- Slope = \( \frac{7}{4} = 1.75 \)
- Y-intercept = \( -\frac{9}{4} = -2.25 \)

Since \( 36 \neq \frac{7}{4} \) and \( -46 \neq -\frac{9}{4} \), the two lines are not the same. Thus, we should find the intersection.

### Step 4: Setting equations equal to find intersection
Now, we set the two equations for \( y \) equal:
\[
36x - 46 = \frac{7}{4}x - \frac{9}{4}
\]
To eliminate the fraction, we can multiply everything by 4:
\[
4(36x - 46) = 7x - 9
\]
\[
144x - 184 = 7x - 9
\]
Now, move terms involving \( x \) to one side and constants to the other:
\[
144x - 7x = 184 - 9
\]
\[
137x = 175
\]
\[
x = \frac{175}{137} \approx 1.286
\]

### Step 5: Finding \( y \)
Substituting \( x \approx 1.286 \) back into either equation to find \( y \). We'll use the first equation:
\[
y = 36(1.286) - 46 \approx 43.296 - 46
\]
\[
y \approx -2.704
\]

### Conclusion:
The lines intersect at approximately \( (1.286, -2.704) \). Since this is a single point of intersection, the solution to the system is a unique intersection point. Therefore, the correct answer from the given options is not explicitly provided, but since they stated both graphs are the same, we also need to analyze options given to confirm:

According to the options provided:
- \((1.286,0)\)
- \(\text{an infinite number of solutions}\)
- \(\text{no solution}\)
- \((1.286,0)\) and \((0,-2.25)\)

None of these points accurately represent the solution derived through the calculations, which were an approximation leading to about \( (1.286, -2.704) \), indicating a unique intersection.

Thus, the correct answer from the given options might be interpreted based on the context of questions, but the accurate solution intersection mathematically is \((1.286, -2.704)\). Since that was not an option and stating “no solution” could imply incorrect options or misunderstanding of identical lines, a revised interpretation suggests they derived them as comparable through informativeness in graph representations rather than exact matching as uniquely identifiable necessarily.

Final answer is:
\[
(1.286,0)\text{ (if understood interchangeably in existence)}; else inconsistently stated with results.
\]

For an exact matching approach depending solely on precise validation, it's not among listed explicitly.