Problem 76 - Prealgebra

Question

Find the value of each of \(12 x-3\) for each of the following values of \(x .\) $$\frac{5}{6}$$

Step-by-Step Solution

Answer

The value of \(12x - 3\) when \(x = \frac{5}{6}\) is 7.

Step 1: Substitute the Value of x

The given expression is \(12x - 3\) and we need to find its value when \(x = \frac{5}{6}\). Substitute \(x = \frac{5}{6}\) into the expression.

Step 2: Perform the Multiplication

Calculate \(12 \times \frac{5}{6}\). Use the rule of multiplying fractions: multiply the numerator by the numerator and the denominator by the denominator. This simplifies to \( \frac{12 \times 5}{6} \).

Step 3: Simplify the Fraction

Simplify \(\frac{12 \times 5}{6}\) to \(\frac{60}{6}\). Simplify the fraction by dividing the numerator by the denominator to get \(10\).

Step 4: Subtraction

Now, substitute back into the original expression to get \(10 - 3\). Perform this subtraction to get \(7\).

Key Concepts

Substitution in Algebra

Substitution in algebra is like replacing ingredients in a recipe. You take a variable, like \(x\), and replace it with a specific number. In our example, we substitute \(x\) with \(\frac{5}{6}\) in the expression \(12x - 3\). This is the initial step in solving many algebra problems, where you get to see what the expression equals by inserting a given number.

• Identify the variable in the expression.
• Replace or "substitute" the variable with the given number.
• Follow through with any necessary calculations.

Substitution helps simplify complex expressions so you can focus on calculations with specific numbers.

Multiplying Fractions

Multiplying fractions may sound tricky, but it is quite straightforward. When you multiply fractions, you follow a simple rule: multiply the numerators and then multiply the denominators.

• Given numbers \(12\) and \(\frac{5}{6}\), treat 12 as \(\frac{12}{1}\).
• Multiply the numerators: \(12 \times 5 = 60\).
• Multiply the denominators: \(1 \times 6 = 6\).

This results in \(\frac{60}{6}\). Don’t worry; once you get the hang of it, multiplying fractions becomes as easy as pie.

Simplifying Fractions

Simplifying fractions means making them easier to understand. A simplified fraction is when you divide the numerator and denominator by their greatest common factor (GCF) until you can't anymore.
In our expression, simplify \(\frac{60}{6}\) by dividing both the numerator and the denominator by 6, which is their GCF.

• Divide the numerator \(60\) by the denominator \(6\).
• Notice that the simplified form is \(10\).

Simplifying fractions is important because it brings the expression to its simplest form, making it easier to understand the resulting value.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operators. They are like math sentences. In \(12x - 3\), \(12x\) suggests we multiply 12 by the variable \(x\), and \(- 3\) tells us to subtract 3 from the product.
Working with expressions includes several steps:

• Identify terms: \(12x\) is a variable term, and \(-3\) is a constant term.
• Apply operations: here, we first multiply and then subtract.
• Achieve a final value once calculations are complete.

Mastering algebraic expressions helps in breaking down complex problems into more manageable parts.