Problem 8 - Intermediate Algebra

Question

Consider the system: \(\left\\{\begin{array}{l}\frac{2}{3} x-\frac{y}{6}=\frac{16}{9} \\ 0.03 x+0.02 y=0.03\end{array}\right.\) a. What algebraic step should be performed to clear the first equation of fractions? b. What algebraic step should be performed to clear the second equation of decimals?

Step-by-Step Solution

Answer

a. Multiply the first equation by 6. b. Multiply the second equation by 100.

Step 1: Identify Least Common Denominator

The first equation has the fractions \( \frac{2}{3} \) and \( \frac{1}{6} \). The least common denominator (LCD) for these fractions is 6.

Step 2: Clear Fractions in First Equation

Multiply every term in the first equation by 6 to eliminate the fractions:\[ 6 \times \left( \frac{2}{3}x \right) - 6 \times \left( \frac{y}{6} \right) = 6 \times \frac{16}{9} \]This simplifies to:\[ 4x - y = \frac{96}{9} \]\[ 4x - y = \frac{32}{3} \]

Step 3: Identify Decimal Adjustment for Second Equation

The second equation involves decimals: 0.03, 0.02, and 0.03. To clear the decimals, multiply the whole equation by 100 to convert decimals into whole numbers.

Step 4: Clear Decimals in Second Equation

Multiply each term in the second equation by 100:\[ 100 \times (0.03x) + 100 \times (0.02y) = 100 \times 0.03 \]This simplifies to:\[ 3x + 2y = 3 \]

Key Concepts

Clearing Fractions

Clearing fractions in an equation is an essential step to simplify the solution process. When you have fractions, dealing with them directly can complicate calculations. Here's how you can make things easier.

Imagine the first equation of our system:

• \[ \frac{2}{3}x - \frac{y}{6} = \frac{16}{9} \]

To clear fractions, multiply every term by the **least common denominator (LCD)** of the fractions involved. By doing this, you eliminate denominators:

• The fractions \( \frac{2}{3} \) and \( \frac{1}{6} \) have a common denominator, which is 6. Multiply the entire equation by 6.

This gives:

• \[ 6 \times \frac{2}{3}x - 6 \times \frac{y}{6} = 6 \times \frac{16}{9} \]

This process results in a simpler equation:

• \[ 4x - y = \frac{32}{3} \]

Clearing fractions makes it much easier to handle equations without worrying about dividing by fractions in subsequent steps.

Clearing Decimals

Dealing with decimals can be daunting, but by transforming them into whole numbers, you simplify the math involved. Let's see how we do this in the second equation in the system:

• \[ 0.03x + 0.02y = 0.03 \]

To clear decimals, decide on a factor that will turn all decimal values into whole numbers. In this case, multiplying by 100 works perfectly. Here's why:

• The smallest decimal place our numbers have is the hundredths, so multiplying by 100 converts these decimals to whole numbers.

This gives:

• \[ 100 \times (0.03x) + 100 \times (0.02y) = 100 \times 0.03 \]

Resulting in:

• \[ 3x + 2y = 3 \]

With this transformation, you eliminate those pesky decimals, making the equation easier to solve by straightforward algebraic methods.

Least Common Denominator

Finding the least common denominator (LCD) is crucial when dealing with fractions in equations. Why is this important? It allows you to eliminate fractions entirely by turning them into whole number coefficients.

Let's take a closer look at how it works:

• Examine the denominators in the equation: \( \frac{2}{3} \,\) and \( \frac{y}{6} \,\).
• Identify the smallest number that is a multiple of all denominators.
• In our case, 6 is a multiple of both 3 and 6, making it the LCD.

Multiply the entire equation by this LCD (i.e., 6) to get rid of the fractions. This adjustment simplifies your equation to:

• \[ 4x - y = \frac{32}{3} \]

By mastering the concept of the LCD, you pave the way for a smoother problem-solving process when dealing with systems of equations involving fractions.

Algebraic Simplification

Algebraic simplification is the process of reducing equations or expressions into their simplest form. This step is vital in making complex equations more manageable and easier to solve.

Here's how it can be applied to our system of equations:

Once you clear out the fractions and decimals, the new equations become:

• \[ 4x - y = \frac{32}{3} \]
• \[ 3x + 2y = 3 \]

Next steps might include:

• Solving the system using substitution or elimination method.
• Simplifying any like terms across either equation.
• Further algebraic manipulation to find the specific values for \(x\) and \(y\).

Ultimately, algebraic simplification prepares your equations for effective solving techniques, ensuring you can address solutions with confidence and clarity.