Problem 64 - Algebra 1

Question

Simplify the expression. $$6\left(-\frac{1}{6} x\right)$$

Step-by-Step Solution

Answer

The simplified expression of \(6\left(-\frac{1}{6} x\right)\) is \(-x\).

Step 1: Identify the numbers to multiply

In the expression \(6\left(-\frac{1}{6} x\right)\), we have a number \(6\) and a fraction \(-\frac{1}{6}\). The operation to be performed here is multiplication. The multiplication operation needs to be done on \(6\) and \(-\frac{1}{6}\). After that we will still keep the variable \(x\) in the expression.

Step 2: Perform the multiplication

Multiply \(6\) by \(-\frac{1}{6}\). Multiplying a whole number by a fraction involves multiplying the whole number by the numerator of the fraction, and dividing it by the denominator of the fraction. In this case, \(6\) multiplied by \(-1\) (which is the numerator of the fraction) is \(-6\). Then we divide \(-6\) by \(6\) (the denominator of the fraction) and we get \(-1\).

Step 3: Write the final simplified expression

Since the multiplication resulted in \(-1\), the final simplified expression is \(-1x\) or simply \(-x\).

Key Concepts

Multiplication of Fractions

Multiplying fractions can initially seem daunting, but it's quite straightforward once you break down the steps. In algebraic expressions, you often encounter fractions interacting with whole numbers or other fractions. To multiply a whole number by a fraction:

• Multiply the whole number by the numerator of the fraction.
• Divide the result by the denominator of the fraction.

For example, in the expression \(6\left(-\frac{1}{6}x\right)\), the multiplication process requires you to multiply \(6\) by \(-1\), resulting in \(-6\).
Then, divide by \(6\), the denominator, which gives you \(-1\). Simple, right?
Remember:

• Keep an eye on the signs (positive or negative) during multiplication.
• Any number multiplied by zero will result in zero.
• Don't forget to bring the variable along with the result!

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. This makes equations easier to understand and solve.
To simplify an expression like \(6(-\frac{1}{6}x)\), follow these steps:

• Complete any multiplication of terms, including constants and fractions.
• Combine like terms if possible. In this exercise, you only simplify the multiplication part as the term \(x\) remains unchanged.

Once the multiplication is done, you're left with \(-1x\).
Dropping coefficients of \(1\) (either positive or negative) gives us the simplest form of \(-x\). Simplifying expressions helps in solving for variables and making further calculations more manageable.

Variables in Algebra

Variables serve as placeholders in algebra, representing unknown or changeable values. They are integral to forming expressions and equations.
In our expression, \(x\) is the variable. When simplifying algebraic expressions:

• Understand that variables can stand alone or be multiplied by coefficients.
• Maintain the variable throughout any mathematical operations unless specified otherwise.
• Remember that simplifying does not change the variable itself, just its coefficient.

In this exercise, \(x\) remains untouched during multiplication; it only affects its coefficient.
Variables allow flexibility and help solve equations for different scenarios, showcasing the power of algebra in diverse situations.