Question

consider the following system of equations 1/3x+y=3, 2y=-2/3x+10 solve using the elimination method show your work

Answer 1

To solve the system of equations using the elimination method, we start with the given equations:

1. \(\frac{1}{3}x + y = 3\) (Equation 1)
2. \(2y = -\frac{2}{3}x + 10\) (Equation 2)

First, let’s rewrite both equations in a more manageable form.

### Step 1: Clear the fractions

For Equation 1, we can multiply through by 3 to eliminate the fraction:

\[
3 \left(\frac{1}{3}x + y\right) = 3 \cdot 3
\]

This simplifies to:

\[
x + 3y = 9 \quad \text{(Equation 3)}
\]

For Equation 2, we already see a way to express it without fractions. We can rearrange it to solve for \(y\):

First, divide the entire equation by 2:

\[
y = -\frac{1}{3}x + 5 \quad \text{(Equation 4)}
\]

### Step 2: Substitute Equation 4 into Equation 3

Now, we'll substitute Equation 4 into Equation 3:

So, substituting \(y\) in Equation 3, we have:

\[
x + 3\left(-\frac{1}{3}x + 5\right) = 9
\]

### Step 3: Simplifying

Expand the equation:

\[
x - x + 15 = 9
\]

This simplifies to:

\[
15 = 9
\]

### Step 4: Analyze the Result

The equation \(15 = 9\) is incorrect and indicates that our system of equations has no solution, implying that the lines represented by these two equations are parallel and do not intersect.

### Conclusion

Thus, the system of equations has no solution. The lines are parallel.