Problem 32 - Algebra 1

Question

Simplify the expression. $$42 y \div \frac{1}{7}$$

Step-by-Step Solution

Answer

The simplified expression is \(294y\).

Step 1: Understand the initial expression

Let's start with the given algebra expression which is \(42y \div \frac{1}{7}\).

Step 2: Recall division by fraction rule

Remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is created by swapping its numerator and the denominator. Thus, the reciprocal of \(\frac{1}{7}\) is \(7\).

Step 3: Apply division by fraction rule

Now apply this rule to the original expression. So the expression \(42y \div \frac{1}{7}\) becomes \(42y * 7\).

Step 4: Simplify the expression

By multiplication, the expression \(42y * 7\) simplifies to \(294y\). That is the simplified version of the original expression.

Key Concepts

Division by a Fraction

When encountering division by a fraction in algebra, it's essential to understand that it's equivalent to multiplying by the inverse or the reciprocal of that fraction. In simple terms, this means if you have a division operation with a fraction in the denominator, you 'flip' the fraction to turn the division into multiplication.

For instance, for the operation \( a \div \frac{b}{c} \), you would multiply \( a \) by the reciprocal of \( \frac{b}{c} \) which is \( \frac{c}{b} \), resulting in \( a \times \frac{c}{b} \). This rule is fundamental in algebra as it simplifies complex expressions and makes them more manageable. When students apply this conversion from division to multiplication, the solving process becomes more intuitive and less error-prone.

Multiplying by the Reciprocal

Multiplying by the reciprocal is a critical concept when working with fractions. If you have an expression where division by a fraction is required, converting this division into multiplication by its reciprocal will simplify your calculations. The reciprocal of a number is what you multiply that number by to get one.

For a fraction \( \frac{x}{y} \), its reciprocal is \( \frac{y}{x} \). For example, to perform the operation \( 10 \div \frac{1}{2} \), we multiply 10 by the reciprocal of \( \frac{1}{2} \) which is 2, leading to \( 10 \times 2 = 20 \). Likewise, in the exercise \( 42y \div \frac{1}{7} \), we identify the reciprocal of \( \frac{1}{7} \) is 7 and multiply \( 42y \times 7 \), simplifying the expression efficiently.

Algebraic Simplification

Algebraic simplification is the process of reducing expressions to their simplest form without changing their value. This process often involves applying a range of operations such as combining like terms, factoring, and using the distributive property. Simplifying can also involve removing parentheses and combining numbers and variables.

For example, the expression \( 3x + 2x - 5 + 3 \) can be simplified by combining like terms to get \( 5x - 2 \). Simplification is invaluable for making complex algebraic expressions understandable and for facilitating further operations like solving equations or inequalities.

Expression Simplification

Expression simplification is crucial in mathematics. It allows us to condense and streamline expressions to make them easier to work with. For instance, when simplifying an expression containing numbers, variables, and operations, you combine terms and utilize arithmetic rules to reduce the expression to a more straightforward and compact form.

In our exercise, \( 42y \times 7 \) simplifies down to \( 294y \), by performing the multiplication operation. A tip for students is to consistently look for opportunities to simplify expressions throughout the solving process. By doing so, you're more likely to avoid mistakes and can solve problems more quickly.