Problem 27 - College Algebra

Question

Solve each system by addition. $$ \begin{array}{l} \frac{5}{6} x+\frac{1}{4} y=0 \\ \frac{1}{8} x-\frac{1}{2} y=-\frac{43}{120} \end{array} $$

Step-by-Step Solution

Answer

\(x = \frac{-43}{215}, y = \frac{86}{129}\).

Step 1: Eliminate Fractions by Finding the Least Common Multiple

To eliminate fractions, find the least common multiple (LCM) of the denominators. For the first equation, the denominators are 6 and 4. The LCM is 12. For the second equation, the denominators are 8, 2, and 120. The LCM is 120. Multiply each term by the LCM to clear fractions.

Step 2: Multiply and Simplify Equations

For the first equation, multiply each term by 12: \(12 \times \frac{5}{6}x + 12 \times \frac{1}{4}y = 12 \times 0\). This simplifies to \(10x + 3y = 0\). For the second equation, multiply each term by 120: \(120 \times \frac{1}{8}x - 120 \times \frac{1}{2}y = 120 \times \frac{-43}{120}\). This simplifies to \(15x - 60y = -43\).

Step 3: Set Up the System with Simplified Equations

The simplified system of equations is:\[ \begin{align*} 10x + 3y &= 0 \ 15x - 60y &= -43 \end{align*} \]

Step 4: Adjust the Equations for Elimination

To eliminate one variable by addition, multiply the first equation by 20 so that the coefficients of \(y\) become equal (with opposite signs): \(20(10x + 3y) = 20(0)\) becomes \(200x + 60y = 0\).

Step 5: Add the Equations to Eliminate \(y\)

Add the modified first equation to the second equation:\[ \begin{align*} 200x + 60y &= 0 \ + \space (15x - 60y &= -43) \end{align*} \]This results in \(215x = -43\).

Step 6: Solve for \(x\)

Solve the equation \(215x = -43\) for \(x\) by dividing both sides by 215:\(x = \frac{-43}{215}\).

Step 7: Substitute \(x\) Back to Solve for \(y\)

Substitute \(x = \frac{-43}{215}\) into the first simplified equation, \(10x + 3y = 0\):\[ 10\left(\frac{-43}{215}\right) + 3y = 0 \]This simplifies to \(-\frac{430}{215} + 3y = 0\). Solving for \(y\) gives \(y = \frac{430}{645} = \frac{86}{129}\).

Key Concepts

Addition Method

The addition method, also called the elimination method, is a useful technique for solving systems of linear equations. This method involves combining two equations to eliminate one of the variables, making it easier to solve for the remaining unknown. In this method, you strategically add or subtract the equations so that one of the variables cancels out. For instance:

• Align the equations vertically with each variable directly above or below its counterpart.
• Adjust the coefficients of one variable so they become equal (but opposite in sign).
• Add the equations to eliminate the chosen variable, reducing the system to a single equation.

By performing these steps, you're left with a simpler equation that includes only one variable, which can then be solved. Once the value of one variable is found, substitute it back into one of the original equations to find the value of the other variable.

Elimination of Fractions

Fractions can complicate algebraic manipulations, especially when solving linear systems. To make calculations more straightforward, it helps to eliminate fractions by using the least common multiple (LCM). This can be done by multiplying each term in an equation by the LCM of its denominators.

When simplifying fractions in linear equations:

• Identify all denominators within the equations.
• Calculate the LCM of these denominators.
• Multiply every term in the equation by the LCM to clear the fractions.

This conversion makes the equation easier to manage and aligns it with standard algebraic processes. For instance, in our system, multiplying by the LCM helped us convert fractional coefficients into whole numbers, which simplified further solving steps.

Least Common Multiple

The least common multiple (LCM) of a set of numbers is the smallest positive integer that is divisible by each number in the set. Finding the LCM is crucial when working with equations containing fractions, as it allows us to eliminate the fractions by scaling.

To find the LCM:

• List the prime factors of each denominator.
• Select the highest power of each prime number that appears in these factorizations.
• Multiply these selected powers together to get the LCM.

In algebra, using the LCM to clear fractions is an effective way to transform the equations into a form where integer manipulation rules apply, thus simplifying the problem-solving process and reducing the chances of errors.

Substitution Method

The substitution method is another common technique for solving systems of equations. It involves solving one equation for a single variable and then substituting that expression into another equation. Although not used primarily in the original exercise, substitution complements elimination methods when an equation is easily solvable for one variable.

Follow these steps to apply the substitution method:

• Rearrange one equation to express one variable in terms of the other.
• Substitute this expression into the other equation, replacing the indicated variable.
• Solve the resulting equation for the remaining variable.
• Back-substitute the found value into one of the original equations to solve for the other variable.

This method is particularly effective when one of the equations is already solved for a variable or can be rearranged easily. Combining substitution with elimination can create a versatile approach to tackling a variety of systems.