Problem 34 - A Graphical Approach to College Algebra

Question

\begin{aligned} &\frac{3}{x}+\frac{2}{y}-\frac{1}{z}=\frac{11}{6}\\\ &\frac{1}{x}-\frac{1}{y}+\frac{3}{z}=-\frac{11}{12}\\\ &\frac{2}{x}+\frac{1}{y}+\frac{1}{z}=\frac{7}{12} \end{aligned}

Step-by-Step Solution

Answer

Solve the system by substitution and elimination to find \( x, y, z \).

Step 1: Identify the Unknowns

We need to find the values of \( \frac{1}{x} = a \), \( \frac{1}{y} = b \), and \( \frac{1}{z} = c \). This transforms the given equation system into a linear system with three variables.

Step 2: Write the Linear System

Re-write the original system using the new variables: \( 3a + 2b - c = \frac{11}{6} \), \( a - b + 3c = -\frac{11}{12} \), \( 2a + b + c = \frac{7}{12} \).

Step 3: Eliminate a Variable

To eliminate \( c \), add the first and second equations: \( 3a + 2b - c + a - b + 3c = \frac{11}{6} - \frac{11}{12} \), simplifying to \( 4a + b + 2c = \frac{11}{12} \).

Step 4: Solve for a Single Variable

Using the new equation and the third equation, \( 2a + b + c = \frac{7}{12} \), subtract the third equation from the result in Step 3: \( 4a + b + 2c - (2a + b + c) = \frac{11}{12} - \frac{7}{12} \), simplifying to \( 2a + c = \frac{4}{12} = \frac{1}{3} \).

Step 5: Substitute Back

Substitute \( c = \frac{1}{3} - 2a \) into one of the original three equations, say \( a - b + 3c = -\frac{11}{12} \), to solve for \( a \) or \( b \).

Step 6: Solve for \( a \) or \( b \)

This will involve replacing \( c \) with \( \frac{1}{3} - 2a \) in one of the equations and solving for the remaining variables using standard algebraic manipulation.

Step 7: Iterate and Resolve Each Variable

Continue substituting and using known values from one equation to another to solve for each variable \( a \), \( b \), and \( c \).

Step 8: Back-substitution for Original Variables

Once solved, use \( a = \frac{1}{x} \), \( b = \frac{1}{y} \), \( c = \frac{1}{z} \) to find \( x, y, z \). For example, \( x = \frac{1}{a} \).

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves the process of transforming equations to make them simpler or to isolate certain variables. In this particular exercise, algebraic manipulation starts with substituting meaningful variables in place of the fractions: \( \frac{1}{x} = a \), \( \frac{1}{y} = b \), and \( \frac{1}{z} = c \). This crucial step simplifies the original non-linear system into a linear one.

Another key point in algebraic manipulation is to consistently perform operations like addition, subtraction, multiplication, or division on both sides of an equation. This helps maintain the equality and ensures that the relationship between variables remains balanced. In Steps 3 and 4 of the solution, the equations are manipulated by adding and subtracting them to eliminate variables and simplify further.

Algebraically manipulating equations by combining and rearranging terms allows us to systematically narrow down our focus to a single variable at a time. This step-by-step simplification process is what ultimately leads us to the solution.

Substitution Method

The substitution method is a technique used to solve systems of equations by replacing one variable with its expression in terms of another variable. This is particularly effective when dealing with linear systems that we simplified through algebraic manipulation.

In this exercise, the substitution method involves making \( c \) an expression of \( a \) and then embedding it into other equations. For instance, from Step 4, we derived \( c = \frac{1}{3} - 2a \). This expression is then substituted into another equation, like \( a - b + 3c = -\frac{11}{12} \), in Step 5, to solve for either \( a \) or \( b \).

The power of substitution lies in its ability to reduce a multi-variable system into a single-variable one. Each substitution step leverages previous results and moves the solution toward a simpler form, eventually solving for all individual variables.

Variable Elimination

Variable elimination is a foundational strategy in solving systems of linear equations. It involves removing one or more variables from the equations, simplifying the system and reducing the number of unknowns.

The elimination process begins in Step 3, where we aim to eliminate \( c \). By adding the first two equations, \( 3a + 2b - c \) and \( a - b + 3c \), the solution systematically consolidates equations, canceling out \( c \) and forming a new equation: \( 4a + b + 2c = \frac{11}{12} \).

This new equation, free of one variable, becomes a crucial tool. As shown in subsequent steps, further elimination is used to simplify the variable mix until the final shake-down to singular equations is possible. By gradually removing variables and creating simpler expressions, variable elimination clarifies the solution path and is essential to handling complex systems.