Problem 95 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the given number is a solution of the equation. $$20-\frac{1}{3} z=\frac{1}{2} z ; 12$$

Step-by-Step Solution

Answer

No, 12 is not a solution to the given equation

Step 1: Substitute the given value into the equation

Replace z with 12 in the equation to get \(20-\frac{1}{3} \cdot 12=\frac{1}{2} \cdot 12\)

Step 2: Simplify each side of the equation

Simplification gives \(20-4=\frac{12}{2}\) which further simplifies to 16=6

Step 3: Compare the two sides of the equation

On comparing, we find that the left hand side does not equal the right hand side (since 16 is not equal to 6). Thus, the number 12 is not a solution to the given equation.

Key Concepts

Algebraic Equations

At the heart of algebra, you'll find algebraic equations—a fundamental concept required to unlock the doors of mathematics. An algebraic equation is a statement of equality between two algebraic expressions that typically involve variables and constants. It represents a scale that has been balanced perfectly. Just as with a physical scale, if you add (or subtract) a weight from one side, you must do the same on the other to keep the balance.

When it comes to solving these equations, the goal is to determine the value of unknown variables that make the equation true. In the exercise given, we are asked whether a specific number (in this case, 12) is the solution to the equation. It's like being a detective: you need to scrutinize the evidence (substitute the number), ponder the facts (simplify the expression), and make a verdict (compare the sides to check for equality). Remember, if the equation holds true with the values substituted for the variables, then you've cracked the case!

Substitution Method

When we face a suspect in algebra—the unknown value—we often use the substitution method to get a confession. It's one of several techniques to find the solution for variables. In the substitution method, you replace the variable with its supposed value and then perform the necessary operations to simplify the expressions. Think of it as swapping out a puzzle piece to see if it fits.

In our exercise, substitution was the first step. The number 12 was offered up as the potential solution for 'z', and we substituted it into the equation. The substitution is like giving us a glimpse of a parallel universe where '12' is indeed 'z', and then we have to verify if this universe obeys the laws of algebraic harmony - does the equation remain balanced or does it tip over?

Simplifying Expressions

To scribble through a jungle of algebraic terms, we need to simplify—a process combing through the underbrush to carve out a clear path. Simplifying expressions means to reduce complex algebraic statements into simpler forms by performing arithmetic operations and combining like terms.

In our test case, once we substituted 12 for 'z', we simplified both sides of the equation systematically. The left side involved arithmetic with fractions, while the right side required dividing 12 by 2. This process isn't just about making the equation look neater; it's essential for properly evaluating each side. By simplifying, we find the purest form of the expression, revealing its true face. When we see clearly what each side equals, it becomes a straightforward task to check if the equation balances or not.