Questions: Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary. 7x-9y=1 -14x+18y=1 A. (x, 1/12 x) B. (x, 2/9+7/9 x) C. (-1/12, 19/108) D. ∅

Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary. 7x-9y=1 -14x+18y=1 A. (x, 1/12 x) B. (x, 2/9+7/9 x) C. (-1/12, 19/108) D. ∅
Transcript text: Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary. \[ \begin{array}{r} 7 x-9 y=1 \\ -14 x+18 y=1 \end{array} \] A. $\left\{\left(x, \frac{1}{12} x\right)\right\}$ B. $\left\{\left(x, \frac{2}{9}+\frac{7}{9} x\right)\right\}$ C. $\left\{\left(-\frac{1}{12}, \frac{19}{108}\right)\right\}$ D. $\varnothing$
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Solution

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Solution Steps

To solve the system of linear equations, we can use the method of elimination or substitution. Here, we will use elimination to determine if the system has a unique solution, infinitely many solutions, or no solution. We will multiply the first equation by 2 and then add it to the second equation to see if we can eliminate one of the variables.

Step 1: Write the System of Equations

We start with the given system of equations: \[ \begin{align*}

  1. & \quad 7x - 9y = 1 \\
  2. & \quad -14x + 18y = 1 \end{align*} \]
Step 2: Analyze the Equations

To determine the relationship between the two equations, we can manipulate them. Notice that the second equation can be rewritten as: \[ -2(7x - 9y) = 1 \] This indicates that the second equation is a multiple of the first equation, but with a different constant on the right side.

Step 3: Check for Consistency

Since the left-hand sides of both equations are proportional (the second equation is \(-2\) times the first), but the right-hand sides are not proportional (1 is not equal to \(-2\)), this indicates that the two lines represented by the equations are parallel and do not intersect.

Step 4: Conclusion

Since the system of equations does not have any points of intersection, we conclude that there is no solution.

Final Answer

The system has no solution, so the answer is \(\boxed{\varnothing}\).

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