Problem 3 - Elementary Algebra

Question

For Problems \(1-12\), solve each of the equations. These equations are the types you will be using in Problems 13-40. $$ \ell+\frac{1}{4} \ell-1=19 $$

Step-by-Step Solution

Answer

The solution is \( \ell = 16 \).

Step 1: Combine Like Terms

The equation is given as \( \ell + \frac{1}{4} \ell - 1 = 19 \). Begin by combining the "\( \ell \)" terms on the left side of the equation. Add \( \ell \) and \( \frac{1}{4} \ell \) to get \( \frac{5}{4} \ell - 1 = 19 \).

Step 2: Isolate the Variable Term

Add 1 to both sides of the equation to move the constant term to the right side. This results in \( \frac{5}{4} \ell = 20 \).

Step 3: Solve for the Variable

To solve for \( \ell \), multiply both sides by the reciprocal of \( \frac{5}{4} \), which is \( \frac{4}{5} \). This gives \( \ell = 20 \times \frac{4}{5} \). Calculate the right side to obtain \( \ell = 16 \).

Key Concepts

Combining Like Terms

Combining like terms is an essential step in simplifying linear equations. When you look at an equation, you often find terms that can be combined. For instance, with the equation \( \ell + \frac{1}{4} \ell - 1 = 19 \), there are two similar terms involving the variable \( \ell \). To simplify, you need to add these like terms together.

• Identify the Like Terms: In the given equation, \( \ell \) and \( \frac{1}{4} \ell \) are like terms since both involve the variable \( \ell \).
• Perform the Addition: To combine them, think of having 1 whole \( \ell \) and an additional \( \frac{1}{4} \) of \( \ell \). When you add these together, you get \( \frac{5}{4} \ell \). This results from the fact that \( 1 + \frac{1}{4} = \frac{5}{4} \).

This step reduces the left side of the equation to \( \frac{5}{4} \ell - 1 \). Simplifying equations this way helps you better see what operations are needed to solve for the variable.

Isolating Variable Terms

After combining like terms, the next important step is to isolate the variable term. This means getting the variable on one side of the equation while moving the constants to the other side. For example, consider the simplified equation from the previous step:

• Move the Constant: Start by removing the \(-1\) from the left side by adding 1 to both sides. This keeps the equation balanced, leading to \( \frac{5}{4} \ell = 20 \).
• Why This Helps: Having the term \( \frac{5}{4} \ell \) isolated makes it easier to see what needs to be done next—specifically what multiplication or division will solve for \( \ell \).

This process of isolating your variable is crucial because it sets you up for solving the equation, allowing for straightforward arithmetic operations.

Multiplication of Fractions

Multiplying fractions is a technique often used to solve equations where a fraction is multiplied by the variable. In the equation we solved, you encounter this operation when isolating \( \ell \). Let's discuss how to handle such situations:

• The Reciprocal: When \( \frac{5}{4} \ell = 20 \), you need to find \( \ell \). To do this, multiply both sides by the reciprocal of \( \frac{5}{4} \), which is \( \frac{4}{5} \). This cancels the fraction attached to \( \ell \).
• Perform the Multiplication: The left side becomes \( \ell \), and the right will be \( 20 \times \frac{4}{5} \). Calculate this by multiplying the numerators and the denominators: \( 20 \cdot 4 = 80 \) and \( 1 \cdot 5 = 5 \), so \( \frac{80}{5} = 16 \).

This approach shows that \( \ell = 16 \). Understanding how to multiply with fractions and apply their reciprocals makes solving equations more intuitive.