Problem 52 - Algebra A Combined Function

Question

The fraction \(\frac{1}{18}\) can be written as the following sum: \(\frac{1}{18}=\frac{x}{2}+\frac{y}{3}+\frac{z}{9}\) where the numbers \(x, y,\) and \(z\) are solutions of \(\left\\{\begin{aligned} x+3 y+z &=-3 \\\\-x+y+2 z &=-14 \\ 3 x+2 y-z &=12 \end{aligned}\right.\) Solve the system and see that the sum of the fractions is \(\frac{1}{18}\).

Step-by-Step Solution

Answer

The solution set is \( x = 1, y = 1, z = -7 \) and it satisfies the sum \( \frac{1}{18} \).

Step 1: Setup the System of Equations

We are given the system of equations: 1. \( x + 3y + z = -3 \)2. \( -x + y + 2z = -14 \)3. \( 3x + 2y - z = 12 \) Our goal is to find values for \( x, y, \) and \( z \) such that these equations hold true.

Step 2: Solve for One Variable

Let's solve the third equation for \( z \):\[ z = 3x + 2y - 12 \] (Equation 3 is rearranged for \( z \)).

Step 3: Substitute into Another Equation

Substitute the expression for \( z \) (from Step 2) into the first equation:\[ x + 3y + (3x + 2y - 12) = -3 \]Simplifying gives:\[ 4x + 5y = 9 \] (Equation 4).

Step 4: Substitute into the Second Equation

Now substitute the expression for \( z \) into the second equation:\[ -x + y + 2(3x + 2y - 12) = -14 \]Simplifying gives:\[ 5x + 5y = 10 \] (Equation 5).

Step 5: Solve System of Two Equations

Solve the system formed by Equations 4 and 5:\( 4x + 5y = 9 \) (Equation 4) and\( 5x + 5y = 10 \) (Equation 5).Subtract Equation 4 from Equation 5 to eliminate \( y \):\[ (5x + 5y) - (4x + 5y) = 10 - 9 \]\[ x = 1 \].

Step 6: Find y

Substitute \( x = 1 \) into Equation 4:\[ 4(1) + 5y = 9 \]\[ 4 + 5y = 9 \]\[ 5y = 5 \]\[ y = 1 \].

Step 7: Find z

Substitute \( x = 1 \) and \( y = 1 \) into the expression for \( z \) from Step 2:\[ z = 3(1) + 2(1) - 12 \]\[ z = 3 + 2 - 12 \]\[ z = -7 \].

Step 8: Verify the Solution

Plug \( x = 1 \), \( y = 1 \), \( z = -7 \) into the expressions of the sum:\[ \frac{x}{2} + \frac{y}{3} + \frac{z}{9} = \frac{1}{2} + \frac{1}{3} - \frac{7}{9} \]To combine, rewrite fractions with a common denominator of 18:\[ \frac{9}{18} + \frac{6}{18} - \frac{14}{18} = \frac{1}{18} \] The sum is indeed \( \frac{1}{18} \).

Key Concepts

Fraction Decomposition

Fraction decomposition is a technique where we express a complex fraction as a sum of simpler fractions. In this exercise, we were tasked with expressing the fraction \( \frac{1}{18} \) as a sum of fractions: \( \frac{x}{2} + \frac{y}{3} + \frac{z}{9} \). This process allows us to break down the fraction into parts that can be individually analyzed and understood. To do this effectively:

• Identify fractions that sum up to the original fraction.
• Ensure that the sum of these fractions matches the original fraction, maintaining the balance.
• Use algebraic methods, such as solving systems of equations, to find unknowns like \( x, y, \) and \( z \).

By breaking down complex fractions, we make them easier to identify and manipulate, aligning with the main goal of solving the system of equations in the given problem.

Substitution Method

The substitution method is a technique used to solve systems of equations by expressing one variable in terms of others. This method is particularly useful when dealing with multiple equations. Here's how it worked in the original exercise:

• From one of the equations, solve for one variable. In this case, we solved for \( z \) from the third equation.
• Substitute this expression into the remaining equations. This helps in reducing the number of variables you are working with in each equation.

This technique simplifies a complex system into more manageable parts, eventually leading to a single equation that can be resolved to find one variable. Repeat the process to solve for all unknowns in the system.

Linear Equations

Linear equations are mathematical expressions that represent straight lines when graphed. These equations have variables raised to the first power. In this problem, the system of equations provided were linear:

• \( x + 3y + z = -3 \)
• \( -x + y + 2z = -14 \)
• \( 3x + 2y - z = 12 \)

The solution to these equations provides the values of \( x, y, \) and \( z \) that satisfy each equation simultaneously. Solving linear equations often involves finding points of intersection in a graph, where all equations are true. This process is essential for various real-world applications, like computing intersection points or balancing quantities.

Solution Verification

Solution verification is a crucial step in ensuring that the answers we have computed are correct. After solving a system of equations, it's necessary to check if the obtained values satisfy the original conditions:

• Substitute the solutions back into the original equations or expressions.
• Ensure all left-hand-side expressions equal their corresponding right-hand sides.

In this exercise, we verified our solutions by substituting \( x = 1 \), \( y = 1 \), and \( z = -7 \) into both the system of equations and the fraction decomposition. The sum indeed simplified back to \( \frac{1}{18} \), confirming the correctness of our work. This step helps in catching potential errors and reinforces the understanding of the problem's requirements.