Problem 31 - Algebra 1

Question

Solve the equation if possible. $$ 24-6 r=6(4-r) $$

Step-by-Step Solution

Answer

The equation is true for all real values of \(r\).

Step 1: Distribute the 6 on the right side of the equation

The right side of the equation is \(6(4-r)\). To simplify, distribute the 6 into the parentheses: \(24 - 6r = 24 - 6r\).

Step 2: Verify if the equation is true

After the distribution in step 1, both sides of the equation became identical, so the original equation is an identity and holds true for any real value of \(r\).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that helps simplify equations and expressions. It's especially useful when you have numbers or variables inside parentheses. Essentially, the distributive property tells us how to multiply a number with a sum or difference inside parentheses.

• It simply states: \( a(b + c) = ab + ac \)
• This means you multiply the number outside the parentheses ("a" in this case) by each of the terms inside the parentheses ("b" and "c").

In the exercise, we applied the distributive property to the expression on the right side of the equation:

• \( 6(4 - r) \) means multiplying 6 by both 4 and \( -r \).
• Thus, \( 6 \times 4 \) equals 24 and \( 6 \times -r \) results in \( -6r \).
• Putting these results together, you get \( 24 - 6r \).

This is how we arrived at the simplified equation: \( 24 - 6r = 24 - 6r \). When using the distributive property, always ensure every term inside the parentheses is multiplied correctly.

Identity Equation

An identity equation is a fascinating concept in mathematics. It’s an equation that is always true, no matter what value you substitute for the variable. This makes it unique because it holds for all numbers, not just specific ones.

• If after simplifying an equation, both sides are identical (like in the original exercise: \( 24 - 6r = 24 - 6r \)), it indicates that the equation is an identity.
• Unlike conditional equations, which are true for specific values of the variable, identity equations are universally true.

Understanding identity equations helps reinforce that sometimes equations are not about finding one solution, but recognizing a fundamental truth about numbers. Identity equations are a marvelous way to see consistency and balance in math.

Real Numbers

Real numbers are the set of all numbers that you would typically use in everyday life. They include:

• Natural numbers (1, 2, 3, ...)
• Whole numbers (0, 1, 2, 3, ...)
• Integers (..., -3, -2, -1, 0, 1, 2, 3, ...)
• Rational numbers (fractions like \( \frac{1}{2} \) or \( 3.5 \))
• Irrational numbers (like \( \pi \) or \( \sqrt{2} \))

Real numbers are continuous and can be found on a number line. For the equation in the exercise, since it’s an identity equation, it holds for any real number substituted for \( r \).This implies our solution isn't limited to just a subset of numbers; it shows the broad application and understanding of real numbers in solving equations. Remember, when working with equations, recognizing the type of numbers involved expands your problem-solving toolkit.