Question

solve the system by the method of your choice.\

$$\begin{cases}3x - 4y = 10\\\\4x - 3y = 4\\end{cases}$$

\select the correct choice below and, if necessary, fill in the answer box to complete your choice.\bigcirc a. there is exactly one solution. the solution set is \boxed{}.\\(\text{(simplify your answer. type an ordered pair.)}\\)\bigcirc b. there are infinitely many solutions. the solution set is \\(\\{(x,y)|3x - 4y = 10\\}\\)or \\(\\{(x,y)|4x - 3y = 4\\}\\).\bigcirc c. the solution set is \\(\varnothing\\).

Explanation:

Step1: Eliminate a variable (e.g., y)

Multiply the first equation by 3: \( 9x - 12y = 30 \)
Multiply the second equation by 4: \( 16x - 12y = 16 \)

Step2: Subtract the equations

Subtract the first new equation from the second: \( (16x - 12y) - (9x - 12y) = 16 - 30 \)
Simplify: \( 7x = -14 \)

Step3: Solve for x

Divide both sides by 7: \( x = \frac{-14}{7} = -2 \)

Step4: Substitute x into an original equation

Use \( 3x - 4y = 10 \), substitute \( x = -2 \): \( 3(-2) - 4y = 10 \)
Simplify: \( -6 - 4y = 10 \)
Add 6 to both sides: \( -4y = 16 \)
Divide by -4: \( y = -4 \)

Answer:

A. There is exactly one solution. The solution set is \((-2, -4)\)