Problem 36 - Algebra A Combined Function

Question

Solve each system of equations by the addition method. If a system contains fractions or decimals, you may want to first clear each equation of fractions or decimals.$$ \left\\{\begin{array}{l} \frac{x}{2}+\frac{y}{4}=1 \\ -\frac{x}{4}-\frac{y}{8}=1 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has no solution; the lines are parallel.

Step 1: Clear Fractions

To eliminate the fractions in the equations, find the least common multiple (LCM) of the denominators, which are 2, 4, and 8. The LCM is 8. Multiply every term in each equation by 8 to clear the fractions.

Step 2: Simplify the Equations

After multiplying by 8, the system becomes: \[4x + 2y = 8\] and \[-2x - y = 8\]. These equations no longer have fractions.

Step 3: Set up for Addition

To use the addition method, we want to eliminate one of the variables by adding the equations. Notice that if we multiply the second equation by 2, the equations will become suitable for elimination of \(y\).

Step 4: Multiply Second Equation

Multiply the second equation \(-2x - y = 8\) by 2 to get \[-4x - 2y = 16\]. Now our system is: \[4x + 2y = 8\] and \[-4x - 2y = 16\].

Step 5: Add the Equations

Add the equations \(4x + 2y = 8\) and \(-4x - 2y = 16\) together: \[(4x + 2y) + (-4x - 2y) = 8 + 16\]. This results in \[0 = 24\], which is a contradiction.

Step 6: Conclude the Solution

Since adding the equations resulted in a false statement, \(0 = 24\), the system of equations has no solution. This signifies the lines represented by these equations are parallel.

Key Concepts

Addition Method

The addition method, also known as the elimination method, is a technique used to solve systems of equations. It involves adding or subtracting the equations in a system to eliminate one of the variables, making it easier to solve for the other variable.
In the system of equations given, the goal was to eliminate the variable 'y' by aligning the coefficients of 'y' in such a way that they would cancel each other out when added. By strategically multiplying and then adding the equations together, we can simplify the system to just one equation with one variable.

• Start by preparing your equations by multiplying them if needed so that one of the variables can be easily eliminated.
• Add the equations together. If done correctly, one of the variables should completely disappear.

This technique is particularly useful because it can turn a complex two-variable problem into a simpler one-variable problem, allowing for a straightforward solution.

Fractions in Equations

Working with fractions in equations can seem daunting at first, but understanding them is essential for solving various algebraic expressions. Fractions commonly appear in real-world problems and can complicate calculations if not handled properly. The presence of fractions necessitates an additional step before proceeding with solving equations.
Firstly, identify all the fractions involved. This may seem trivial, but clearly recognizing your fractions is the base step.

• The next crucial step is to eliminate the fractions by finding a common multiple for all the denominators involved. This process simplifies the equations drastically.
• Multiply every term in the equation by the least common multiple (LCM) of the denominators. This action clears the fractions, transforming the equation into a more manageable form with whole numbers.

Removing fractions simplifies the computational process significantly, enabling easier application of the addition method.

Eliminating Fractions

Eliminating fractions from a system of equations is a valuable technique, as it simplifies the process by converting the equations into a more intuitive and elementary form. This technique involves using the least common multiple (LCM) of the denominators of all fractions present.
To start, calculate the LCM of all the denominators present in your system. Once you've determined this, multiply each term of every equation by this LCM. This procedure will convert each fraction into an integer.

• By clearing the fractions, you essentially transform the painstaking task of working with fractions into a simpler task of managing whole numbers.
• The resulting system of equations is typically easier to manipulate, making it straightforward to use techniques like the addition method to find a solution.

Keep in mind that eliminating fractions doesn't change the solutions of the system. It merely makes the arithmetic more accessible, allowing for a smoother solving process.