Question

Consider the following system of equations:
1/3x+y=3
2y=-2/3x+10
A. Is (0, 3) a solution to the system of equations? Explain.

Answer 1

Solve the system of equations using the elimination method.

Answer 2

To determine if the point (0, 3) is a solution to the system of equations, we need to substitute \( x = 0 \) and \( y = 3 \) into both equations and check if they hold true.

First, let's take the first equation:
\[
\frac{1}{3}x + y = 3
\]
Substituting \( x = 0 \) and \( y = 3 \):
\[
\frac{1}{3}(0) + 3 = 3
\]
This simplifies to:
\[
0 + 3 = 3
\]
which is true.

Now, let's check the second equation:
\[
2y = -\frac{2}{3}x + 10
\]
Substituting \( x = 0 \) and \( y = 3 \):
\[
2(3) = -\frac{2}{3}(0) + 10
\]
This simplifies to:
\[
6 = 0 + 10
\]
This is not true, as \( 6 \) does not equal \( 10 \).

Since (0, 3) satisfies the first equation but not the second, we conclude that:
**(0, 3) is not a solution to the system of equations.**

Answer 3

To solve the system of equations using the elimination method, we first rewrite the equations for clarity:

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

We'll first eliminate fractions for easier calculations.

### Step 1: Clear fractions from both equations.
For the first equation:
\[
\frac{1}{3}x + y = 3
\]
Multiply through by 3:
\[
x + 3y = 9 \quad \text{(Equation 1)}
\]

For the second equation:
\[
2y = -\frac{2}{3}x + 10
\]
Multiply through by 3 to eliminate the fraction:
\[
6y = -2x + 30
\]
Now, rearranging gives:
\[
2x + 6y = 30 \quad \text{(Equation 2)}
\]

### Step 2: Align the equations.
Now we have:
1. \( x + 3y = 9 \)
2. \( 2x + 6y = 30 \)

### Step 3: Prepare for elimination.
Next, we notice that the second equation can be simplified. Dividing all terms in Equation 2 by 2 gives:
\[
x + 3y = 15 \quad \text{(Equation 3)}
\]

### Step 4: Set up the system for elimination.
Now we have:
1. \( x + 3y = 9 \) (Equation 1)
2. \( x + 3y = 15 \) (Equation 3)

### Step 5: Subtract the equations.
Subtract Equation 1 from Equation 3:
\[
(x + 3y) - (x + 3y) = 15 - 9
\]
This simplifies to:
\[
0 = 6
\]

### Conclusion.
This result, \(0 = 6\), indicates that the system of equations is inconsistent. Therefore, the system has no solution.

The two equations represent parallel lines that do not intersect.