Problem 29 - Algebra 1

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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: \[ \begin{aligned} &u + 2v = 150 \ &u - 2v = 150 \end{aligned} \]

Step 2: 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 \( u = 150 \).

Step 3: 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 \( v = -150 \).

Step 4: 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.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations that involve the same set of variables. The goal is to find values for these variables that make all the given equations true simultaneously. In our original exercise, we have a system consisting of two equations:

• \(2v = 150 - u\)
• \(2u = 150 - v\)

These equations are related as they both involve the variables \(u\) and \(v\). To find the solution, we aim to determine the specific values for \(u\) and \(v\) that satisfy both equations at the same time.
Solving such systems involves various methods, one popular approach being linear combinations or the addition method. This approach helps eliminate one of the variables by combining the equations, leading to an easier solution path.

Linear Equations

Linear equations are equations of the first degree, meaning they involve only the option of variables raised to the power of one. In their most basic form, linear equations take the shape \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) and \(y\) are variables.
In the context of solving a system of linear equations, putting both equations in a similar format helps to apply solving techniques effectively. For instance, in our problem, rearranging the original equations results in:

• \(u + 2v = 150\)
• \(u - 2v = 150\)

With both equations matched in format, it simplifies the choice to apply the linear combination method as portions of the equation easily cancel out.

Solution Verification

Solution verification is a crucial step in mathematical problem-solving, especially when solving systems of equations. After determining a solution, it's important to check if it holds true for all original equations in the system.
In our exercise's solution, although calculations led to the results \(u = 150\) and \(v = -150\), substitution back into the original equations revealed discrepancies. This step is vital as it ensures that no mistakes were made in solving the equations, such as errors in algebraic manipulation.
When verifying:

• Substitute solutions back into the original equations.
• Check if both sides of the equations are equal after substitution.
• Revisit and reassess your approach if any contradictions are noticed, like an identity not holding true (e.g., \(-300 eq 0\)).

Solution verification allows us to confidently confirm whether the obtained solution is accurate or pinpoint where modifications are necessary.