Question

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{r}2 u+v=120 \\ u+2 v=120\end{array}\right.$$

Short answer

The solution to the system is \(u = 40\) and \(v = 40\). The system is consistent because a unique solution exists.

Step-by-step solution

Multiply equations

To eliminate one of the variables, we need to make coefficients the same. Multiply the first equation by 2 and the second equation by 1. The system becomes: \(4u + 2v = 240\) and \(u + 2v = 120\).

Subtract equations

Subtract the second equation from the first one, the result is \(3u = 120\).

Solve for u

To find the value of \(u\), divide both sides of the equation by 3. The result is \(u = 40\).

Substitution

Substitute \(u = 40\) into the second equation from the original system to find the value for \(v\). Therefore, \(40 + 2v = 120\). To isolate \(v\), subtract 40 from both sides and then divide by 2, giving \(v = 40\).

Check solution

Substitute \(u = 40\) and \(v = 40\) into the original equations to check if they are correct. Both equations are satisfied, hence the solution is correct.

Determine the system consistency

Since we found a unique solution, the system is consistent.

Key concepts

Elimination Method

The elimination method is a popular technique for solving systems of linear equations. It works by eliminating one variable to simplify the system. You'll be tackling two equations and aiming to transform them into one simple equation with a single variable.

Here's a step-by-step approach to use this method efficiently:

• Start by arranging both equations in standard form: aligning all terms nicely on one side of the equation while zeroing out the right-hand side.
• Identify a variable to eliminate. Equalize the coefficients of this variable in both equations by carefully choosing which equations to multiply.
• Once the coefficients match, either add or subtract the equations. This will remove one variable, giving you an equation with one unknown.
• Solve the resulting equation to find the value of the unknown variable.
• Back-substitute this value into one of the original equations to find the other variable.
• Finally, substitute both values back into both original equations to check correctness. Getting both equations satisfied is crucial to verifying your solution.

This method is efficient and clear, often leading you directly to the correct solution, assuming you've made careful calculations along the way.

Consistent System

A consistent system in mathematics refers to a set of equations that have at least one solution. When solving a system of linear equations, it's crucial to determine if the system is consistent.

Here's how you establish the consistency or inconsistency of a system:

• Upon solving the equations, if you find a solution that satisfies all equations, you have a consistent system. For instance, the exercise's solution of \( u = 40 \) and \( v = 40 \) rendered both equations true, confirming consistency.
• If you discover no such solution because the equations lead to a contradiction (such as equal coefficients creating different constants), the system is inconsistent, meaning it has no solutions.

Consistent systems can be further classified into specific categories:

• **Unique Solution:** Just one specific set of values solves the equations, much like the unique solution in the exercise \( (u, v) = (40, 40) \).
• **Infinite Solutions:** Occurs when equations describe the same line, giving rise to an infinite range of solutions.

Understanding this concept is crucial as it helps determine the feasibility of finding a solution for the problem you're attempting to solve.

Linear Equations

Linear equations form the backbone of the exercise at hand. These are algebraic expressions where each term is linear, meaning the highest power of the variable involved is one.

Key aspects of linear equations include:

• **Form:** Typically expressed in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables.
• **Graphical Representation:** When plotted, linear equations result in straight lines. Their intersection points, if any, often represent solutions to the system of equations.

Linear equations are used in various applications from physics to economics because they represent real-world relationships in a simplified linear form.

In our exercise, we had two linear equations: \( 2u + v = 120 \) and \( u + 2v = 120 \). Solving them gives a clear idea of how variables relate to one another in a linear system, providing insights into problem-solving strategies.