Problem 38 - Algebra Form and Function

Question

Solve for \(L\) : $$ \frac{3 k L-8}{r k L-7}=5 $$ Your answer may involve other letters.

Step-by-Step Solution

Answer

Question: Solve for L in the equation $\frac{3kL - 8}{rkL - 7} = 5$. Answer: $L = \frac{27}{(5r-3)k}$.

Step 1: Clear fractions

To eliminate the fraction, multiply both sides of the equation by the denominator, which is \((rkL - 7)\): $$ (rkL - 7)\left(\frac{3kL - 8}{rkL - 7}\right) = 5(rkL - 7) $$

Step 2: Simplify the equation

Now, simplify by canceling the \((rkL - 7)\) on the left side and distribute the \(5\) on the right side: $$ 3kL - 8 = 5rkL - 35 $$

Step 3: Move terms with L to one side of the equation

To isolate L, let's first move all terms with \(L\) to one side of the equation. Subtract \(3kL\) from both sides: $$ -8 = 5rkL - 3kL - 35 $$

Step 4: Combine L terms and isolate L

Combine the L terms and then add \(35\) to both sides: $$ -8+35 = (5r-3)kL $$ Now, divide by the coefficient of \(L\) to isolate \(L\): $$ L = \frac{27}{(5r-3)k} $$

Step 5: Write down the solution

We have now found the expression for \(L\) in terms of the other variables, \(k\) and \(r\): $$ L = \frac{27}{(5r-3)k} $$

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of unknown variables. An equation is a mathematical statement where two expressions are equal. To solve an equation, you perform operations that simplify the equation and isolate the variable to find its value. Sometimes, equations can involve fractions, requiring specific steps to clear them out.

Here are some key steps in solving equations:

• Identify the equation and the variable that needs to be isolated.
• Perform operations such as addition, subtraction, multiplication, or division to simplify the equation.
• When dealing with equations with fractions, find a common denominator or multiply both sides by the denominator to eliminate the fractions.
• Perform the same operations on both sides of the equation to maintain balance.

Through these steps, you will isolate the variable and determine its value, as you have seen by solving for \( L \) in the given exercise.

Rational Equations

Rational equations are equations that involve fractions made up of polynomials in the numerator and denominator. Solving these types of equations usually requires clearing fractions to simplify the problem. When you multiply each term by the least common denominator, it eliminates the fractions, allowing the equation to become simpler.

Here is how you manage rational equations step-by-step:

• Identify the least common denominator (LCD) if multiple terms are present.
• Multiply every term in the equation by the LCD to cancel the fractions.
• Solve the resulting equation as if it's a regular linear equation.
• Check for extraneous solutions, which may arise when multiplying both sides by terms involving the variable.

Understanding rational equations and how to manipulate them is essential, especially for handling complex algebraic expressions elegantly.

Variable Isolation

Variable isolation is the process of manipulating an equation in such a way that the variable you want to solve for stands alone on one side of the equation. This is often the end goal when solving equations. Isolating a variable involves moving all other terms to the opposite side of the equation, allowing you to clearly see the solution.

Here are some tips to successfully isolate a variable:

• Add, subtract, multiply, or divide both sides of the equation by the same number to isolate the variable.
• Carefully combine like terms, especially when there are multiple variables involved.
• Rearrange the terms to place the variable alone on one side of the equation.
• Continue simplifying the equation until the variable is isolated.

Mastering variable isolation empowers you to solve equations with confidence, as shown in isolating \( L \) from the equation \( \frac{3 k L-8}{r k L-7}=5 \).