Problem 51 - Algebra 1

Question

Decide which variable to eliminate when using linear combinations to solve the system. Explain your thinking. $$\begin{array}{l}\frac{1}{3} x+6 y=6 \\\\-x+3 y=3\end{array}$$

Step-by-Step Solution

Answer

The variable chosen to eliminate was x. After elimination, the solution to the system is x = 0, y = 1.

Step 1: Multiply the Equations

To eliminate the x-variable, both equations should be multiplied such that the coefficients of x are opposites. Therefore, multiply the first equation by 3 and the second by 1. The equations become: \\(1) \: x + 18y = 18 \\(2) -x + 3y = 3

Step 2: Addition of Modified Equations

Add equation (1) and equation (2), which results in the cancellation of the x-variable by adding matching terms: \\0 + 21y = 21. This simplifies to y = 1.

Step 3: Solving for Remaining Variable

Now substitute y = 1 into one of the original equations. It's simpler to use the second equation, -x + 3y = 3, for this. With y = 1, the equation becomes -x + 3 = 3, which simplifies to x = 0.

Key Concepts

Understanding System of Equations

A system of equations is a group of two or more equations with unknown variables. The goal is usually to find the values of these variables that satisfy all the equations simultaneously. For example, in a system of equations involving two variables, say \(x\) and \(y\), you have equations like \(\frac{1}{3}x + 6y = 6\) and \(-x + 3y = 3\).

• Each equation provides a relational expression between the unknowns.
• Finding a solution means identifying the point(s) where these equations intersect when plotted.
• Many real-world situations, such as optimization problems and budgeting, can be modeled as a system of equations.

Learning to handle these systems is crucial for deeper mathematical understanding and practical applications.

The Role of Variable Elimination

Variable elimination is a method to simplify solving a system of equations. It's achieved by adding or subtracting multiple equations to remove one of the unknowns, making it easier to solve for the remaining variable.Consider the system given as an example, where you have chosen to eliminate the \(x\)-variable. This decision was based on ease, as aligning the coefficients to be opposites simplifies arithmetic. By manipulating the equations:

• Multiply the first equation by 3 so that the coefficient of \(x\) matches the opposite of the coefficient in the second equation.
• After multiplication, add both equations together; the \(x\) terms cancel out to zero, leaving a simple equation with only \(y\) remain.

Variable elimination reduces the system to a single-variable equation, a stepping stone to finding the complete solution.

Solving Linear Equations Simplified

Solving linear equations involves finding the value of an unknown variable that makes the equation true. In the context of a system, after reducing the system with variable elimination, you solve what is left.Here's how it happens:

• Once we've eliminated \(x\), the modified equation becomes \(21y = 21\).
• From here, simply divide both sides by 21 to solve for \(y\).
• Substitute this solution back into one of the original equations to solve for \(x\).

In our example, substituting \(y = 1\) in the second equation \(-x + 3y = 3\), gives us \(-x + 3 = 3\). Solving for \(x\), we find \(x = 0\).The process demonstrates an effective way to break down and conquer the challenges posed by systems of linear equations.