Problem 33 - Algebra A Combined Function

Question

Solve each system of equations by the substitution method. $$ \left\\{\begin{array}{l} \frac{1}{3} x-y=2 \\ x-3 y=6 \end{array}\right. $$

Step-by-Step Solution

Answer

Infinite solutions; any \( (x, y) \) where \( x = 3y + 6 \).

Step 1: Solve for x in the second equation

The second equation is \( x - 3y = 6 \). We can solve for \( x \) by adding \( 3y \) to both sides:\[ x = 3y + 6 \]

Step 2: Substitute x in the first equation

Now that we have \( x = 3y + 6 \), we substitute this expression into the first equation.The first equation is \( \frac{1}{3}x - y = 2 \). Replacing \( x \) gives:\[ \frac{1}{3}(3y + 6) - y = 2 \]

Step 3: Simplify the first equation

First, distribute \( \frac{1}{3} \) across \( (3y + 6) \):\[ y + 2 - y = 2 \]

Step 4: Solve for y

Simplify the equation:\[ 2 = 2 \]This simplification shows that the equation is always true, indicating that there is no unique solution for \( y \) based on this equation alone.

Step 5: Find x for different values of y

From Step 1, \( x = 3y + 6 \). Since the equation in Step 4 is always true, \( y \) can take any value, and \( x \) is dependent on \( y \). For example, if \( y = 0 \), then \( x = 6 \). More generally, \( x \) and \( y \) are linearly dependent.

Key Concepts

System of Equations

When dealing with a system of equations, you are essentially working with two or more equations that share variables. The goal is to find values for those variables that satisfy all the equations simultaneously.
In the problem given, we have two equations: \( \frac{1}{3} x - y = 2 \) and \( x - 3y = 6 \). These equations are connected through their variables \( x \) and \( y \):

• The first equation is a fraction-based expression involving both variables.
• The second equation has a simpler linear format and can be directly solved for one variable.

This system probably intersects at an infinite number of points which means the two lines lie on top of each other on a graph. Solving such systems involves understanding whether there is one solution, no solution, or infinitely many solutions.

Linear Equations

Linear equations are the building blocks of a system of equations, like the one in our example. A linear equation in two variables represents a straight line on a coordinate plane.
Let's break down what we have:
The first equation, \( \frac{1}{3} x - y = 2 \), is a linear equation where \( x \) and \( y \) contribute to a straight line.
The second line is clear in \( x - 3y = 6 \), representing another different straight line in the plane.
Characteristics of linear equations include:

• Having constant coefficients and variables raised strictly to the first power.
• Typically represented in the form \( Ax + By = C \).
• When graphed, they form lines that might intersect each other at a point or run parallel or even coincide.

Linear dependencies or intersections lead us to possible solutions or dependencies between the variables.

Solving Equations

Solving equations involves finding values for your unknowns—here, \( x \) and \( y \). In systems, we aim to solve each equation such that they hold true simultaneously.
Using the substitution method, we solve one equation for one variable and substitute back into the other to find the second variable's value:
For example, from the equation \( x - 3y = 6 \), we solve for \( x \):

• \( x = 3y + 6 \)

We substitute \( x = 3y + 6 \) into the first equation \( \frac{1}{3}(3y + 6) - y = 2 \).
This simplifies and results in \( 2 = 2 \), implying that the equations are dependent. In this case, the equations describe the same line, leading to many solutions. Here, \( y \) can be any real number, and \( x \) will automatically adjust to \( 3y + 6 \).
Thus,

• Solving involves understanding whether equations are independent, identical, or something in between.
• For identical or dependent lines, we have infinite solutions, as is the case here.

This highlights the importance of interpreting solutions correctly.