Questions: Solve by using substitution or elimination by addition. 4x - 3y = 23 3x + 2y = 13 Select the correct choice below and fill in the answer box within your choice A. The unique solution to the system is B. There are an infinite number of solutions. (Type an ordered pair.) C. There is no solution.

Solution Steps

To solve the system of linear equations using substitution or elimination, we can follow these steps:

1. Use the elimination method to eliminate one of the variables by adding or subtracting the equations.
2. Solve for the remaining variable.
3. Substitute the value of the solved variable back into one of the original equations to find the value of the other variable.
4. Check the solution by substituting both values back into the original equations to ensure they satisfy both equations.

We start with the given system of equations: \[ \begin{align_} 4x - 3y &= 23 \quad (1) \\ 3x + 2y &= 13 \quad (2) \end{align_} \]

To eliminate one of the variables, we can multiply equation (1) by 2 and equation (2) by 3: \[ \begin{align_} 2(4x - 3y) &= 2(23) \\ 3(3x + 2y) &= 3(13) \end{align_} \] This gives us: \[ \begin{align_} 8x - 6y &= 46 \quad (3) \\ 9x + 6y &= 39 \quad (4) \end{align_} \]

Now, we add equations (3) and (4) to eliminate \(y\): \[ (8x - 6y) + (9x + 6y) = 46 + 39 \] This simplifies to: \[ 17x = 85 \]

Dividing both sides by 17, we find: \[ x = 5 \]

Now, we substitute \(x = 5\) back into one of the original equations, say equation (1): \[ 4(5) - 3y = 23 \] This simplifies to: \[ 20 - 3y = 23 \] Rearranging gives: \[ -3y = 3 \quad \Rightarrow \quad y = -1 \]

Final Answer