Questions: Solve by the elimination method. Also, determine whether the system is consistent or inconsistent, and whether the equations are dependent or independent. Use a graphing calculator to check your answer. 1/7 x + 1/3 y = 4 2/7 x - 1/3 y = 2 (Hint: First multiply by the least common denominator to clear fractions.) Select the correct choice below and, if necessary, fill in the answer box within your choice. A. There is one solution. The solution is . (Simplify your answer. Type an ordered pair.) B. There are infinitely many solutions of the form ( x , ). (Simplify your answer.) C. There is no solution.

Solution Steps

To eliminate the fractions, multiply both equations by the least common denominator (LCD) of the coefficients. The denominators in the first equation are 7 and 3, so the LCD is 21. Multiply the first equation by 21: \[ 21 \left( \frac{1}{7}x + \frac{1}{3}y \right) = 21 \cdot 4 \] Simplify: \[ 3x + 7y = 84 \]

Similarly, multiply the second equation by 21: \[ 21 \left( \frac{2}{7}x - \frac{1}{3}y \right) = 21 \cdot 2 \] Simplify: \[ 6x - 7y = 42 \]

Now, the system of equations is: \[ \begin{cases} 3x + 7y = 84 \\ 6x - 7y = 42 \end{cases} \]

Add the two equations to eliminate \( y \): \[ (3x + 7y) + (6x - 7y) = 84 + 42 \] Simplify: \[ 9x = 126 \] Solve for \( x \): \[ x = \frac{126}{9} = 14 \]

Substitute \( x = 14 \) into the first equation to solve for \( y \): \[ 3(14) + 7y = 84 \] Simplify: \[ 42 + 7y = 84 \] Subtract 42 from both sides: \[ 7y = 42 \] Solve for \( y \): \[ y = \frac{42}{7} = 6 \]

The system has a unique solution \( (x, y) = (14, 6) \). This means:

• The system is consistent (it has at least one solution).
• The equations are independent (they are not multiples of each other).

Final Answer