Question

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

Answer 1

To find the solution to the equations given, we will first rewrite both equations in standard form.

### Equation 1:
Starting with:
\[
\frac{7}{2}x - \frac{9}{2} = 2y
\]
We can solve for \(y\) by isolating it:
\[
2y = \frac{7}{2}x - \frac{9}{2}
\]
Dividing every term by 2:
\[
y = \frac{7}{4}x - \frac{9}{4}
\]

### Equation 2:
Now, let’s rewrite the second equation:
\[
4y = 7x - 9
\]
We can solve for \(y\) by dividing every term by 4:
\[
y = \frac{7}{4}x - \frac{9}{4}
\]

Both equations simplify to:
\[
y = \frac{7}{4}x - \frac{9}{4}
\]

### Finding the Solution
Next, we can determine the intersection point of the two lines, which will also be the solution to the equations.

We already see that both equations are equivalent, meaning they represent the same line. This means they will have an infinite number of solutions, as they overlap completely.

However, if we want specific values for a variable, we can choose any value for \(x\) and then find the corresponding \(y\).

#### Example:
Let’s set \(x = 0\):
\[
y = \frac{7}{4}(0) - \frac{9}{4} = -\frac{9}{4}
\]
Thus, one solution is:
\[
(0, -\frac{9}{4})
\]

Therefore, the solution of the system of equations \(\frac{7}{2}x - \frac{9}{2} = 2y\) and \(4y = 7x - 9\) is not a unique point; instead, they represent the same line and thus have infinitely many solutions, all of which satisfy \(y = \frac{7}{4}x - \frac{9}{4}\).