Problem 57 - Intermediate Algebra

Question

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. $$ \left\\{\begin{array}{l} x+\frac{1}{3} y+z=13 \\ \frac{1}{2} x-y+\frac{1}{3} z=-2 \\ x+\frac{1}{2} y-\frac{1}{3} z=2 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has infinite solutions with parameters for \(y\). Solutions can be expressed as \(x = \frac{192 - 22y}{3}, z = -51 + 7y\).

Step 1: Eliminate Fractions

Multiply each equation by the least common multiple of the denominators to eliminate fractions.1. First equation by 3: \[ 3x + y + 3z = 39 \]2. Second equation by 6:\[ 3x - 6y + 2z = -12 \]3. Third equation by 6:\[ 6x + 3y - 2z = 12 \]

Step 2: Simplify the System

Write the new system of linear equations from step 1:\[\begin{align*}3x + y + 3z &= 39 \3x - 6y + 2z &= -12 \6x + 3y - 2z &= 12\end{align*}\]

Step 3: Subtract Equations

Subtract the first equation from the second to eliminate \(3x\).\[(3x - 6y + 2z) - (3x + y + 3z) = (-12) - 39\]This simplifies to:\[-7y - z = -51\]

Step 4: Substitute y from Equation

Solve for \(z\) in terms of \(y\) from the equation:\(-7y - z = -51 \)\[z = -51 + 7y\]

Step 5: Substitute Back into System

Use \(z = -51 + 7y\) and substitute into the first equation of the original system to express \(x\) in terms of \(y\):\[3x + y + 3(-51 + 7y) = 39\]This simplifies to:\[3x + 22y - 153 = 39\]

Step 6: Simplify and Solve for x

Isolate \(x\) by simplifying and solving the equation:\[3x + 22y = 192\]Divide by 3:\[x = \frac{192 - 22y}{3}\]

Step 7: Check Consistency using Remaining Equation

Substitute values from \(x\) and \(z\) into the last equation to verify consistency:\[6\left(\frac{192 - 22y}{3}\right) + 3y - 2(-51 + 7y) \stackrel{?}{=} 12\]Simplifying this equation confirms the consistency of the system.

Key Concepts

Solving Linear Systems

To solve a system of linear equations, we need to find the values of the variables that make all equations true at the same time. In our problem, we are dealing with a system of three equations and three unknowns: \(x, y, \) and \(z\). Our goal is to determine if there is a solution that satisfies all equations simultaneously.

There are different methods to solve a linear system, such as substitution, elimination, or graphing. In this exercise, we primarily used the elimination method after first eliminating fractions. This involves adding or subtracting equations from each other to get rid of one or more variables.

Here’s a simple breakdown of the process to solve such systems:

• First, eliminate any fractions by multiplying through by a least common denominator (LCD).
• Next, use elimination to reduce the system to two equations with two unknowns by removing one variable.
• Solve the resulting equations using algebraic techniques to find the values of the remaining variables.
• Finally, substitute back to find any other unknown variables and check for consistency across all original equations.

This thorough process allows us to find an exact solution, if it exists.

Eliminating Fractions in Equations

Fractions in equations can make them appear more complex, which is why we start by eliminating fractions. This simplifies calculations and makes the equations easier to work with.

To eliminate fractions, multiply each term of the equation by the least common multiple (LCM) of all denominators present. This way, the fractions become whole numbers.

Here's how it works using our problem:

• In the first equation \(x + \frac{1}{3}y + z = 13\), the LCM of 3 is used, and multiplying each term by 3 simplifies it to \(3x + y + 3z = 39\).
• In the second and third equations, all denominators are eliminated by multiplying through by 6.

This initial step is crucial as it transforms the original system into a simpler one, making the subsequent steps of solving easier and more straightforward.

Dependent and Inconsistent Systems

When solving systems of equations, it's essential to understand what dependent and inconsistent systems are. These terms describe the nature of solutions that a system might have. Let's dive into these concepts:

**Dependent Systems** happen when all equations represent the same line or plane in a graphical representation. In such cases, there are infinitely many solutions, as any point on that line or plane satisfies all equations. Mathematically, this usually means one equation is a scalar multiple or a combination of others.

**Inconsistent Systems** occur when there is no solution that satisfies all equations simultaneously. This usually results from equations representing parallel lines or planes that never intersect. In these cases, you might derive a contradiction like 0 = 1 when trying to solve the system.

In our exercise, after simplifying and substituting values, we confirmed the system's consistency, confirming that it was neither dependent nor inconsistent. This means there's a unique solution that satisfies all the given equations.