Consider the following system of equations:

Question

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

Answer 1

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 2

Solve the system of equations using the elimination method.

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.