Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &2 v=150-u\\\&2 u=150-v\end{aligned}$$

Short answer

The solution \( u = 150 \) and \( v = -150 \) does not apply to the system of equations. The system has no solution.

Step-by-step solution

Rearrange the Equations

The first step towards solving this problem is to rearrange both equations to place variables and constants on their respective sides. After rearranging, the equations could look like: \[ \begin{aligned} &u + 2v = 150 \ &u - 2v = 150 \end{aligned} \]

Apply Linear Combination Method

Use the addition method or linear combination to eliminate one of the variables. Add the two equations together: \( (u + 2v) + (u - 2v) = 150 + 150 \), which simplifies to \( 2u = 300 \). Then, divide both sides by 2 to get \( u = 150 \).

Substitute \( u \) into Second Equation

Substitute the value of \( u \) from step 2 into the second equation: \(2u = 150 - v \). This gives \( 2 * 150 = 150 - v \), which simplifies to \( 300 = 150 - v \). By rearranging this equation, we find that \( v = -150 \).

Verification

Finally, for verification purposes substitute \( u \) and \( v \) into the original system of equations to check if the system holds true. \[ \begin{aligned} &2 * (-150) = 150 - 150 \ &2 * 150 = 150 + 150 \end{aligned} \] The equations become \( -300 = 0 \) and \( 300 = 300 \), thus the first equation does not apply, our solution must be false.