Problem 78 - Elementary Algebra

Question

\(18\left(\frac{5}{6} h-\frac{1}{3}\right)-7\)

Step-by-Step Solution

Answer

15h - 13

Step 1 - Distribute the 18

Use the distributive property to multiply 18 by each term inside the parentheses: \[ 18 \times \frac{5}{6} h - 18 \times \frac{1}{3} \]

Step 2 - Simplify the expressions

Simplify each product inside the parentheses: \[ 18 \times \frac{5}{6} h = 15h \] and \[ 18 \times \frac{1}{3} = 6 \]

Step 3 - Substitute back into the expression

Substitute the simplified terms back into the expression: \[ 15h - 6 \]

Step 4 - Subtract 7

Subtract 7 from the simplified expression: \[ 15h - 6 - 7 \]

Step 5 - Combine like terms

Combine the constants to get the final simplified expression: \[ 15h - 13 \]

Key Concepts

Simplifying Expressions

In algebra, simplifying expressions is a fundamental skill.
It makes equations easier to work with and understand.
When simplifying, you want to reduce an equation to its simplest form.
This often includes distributing terms, combining like terms, and eliminating unnecessary elements. For instance, consider the expression provided: \(18\left(\frac{5}{6}h-\frac{1}{3}\right)-7\).
The first step involves using the distributive property, which means multiplying each term inside the parentheses by 18.
This gives us \( 18 \times \frac{5}{6}h - 18 \times \frac{1}{3} \).

Next, we simplify these products: \( 18 \times \frac{5}{6} h = 15h \) and \( 18 \times \frac{1}{3} = 6 \).
Substituting these back into the expression results in \( 15h - 6 \).
Such steps simplify the overall equation, making it much more manageable.

Combining Like Terms

Combining like terms is another crucial step in simplifying expressions.
Like terms refer to terms that have the same variable raised to the same power.
In our example, we see that after simplification, we get \(15h - 6 - 7\).
Here, both -6 and -7 are constants. Combining like terms means adding or subtracting these constants, while leaving the variable term intact.
So, we combine -6 and -7: \(-6 - 7 = -13\).
As a result, our final expression becomes \(15h - 13\).
By combining like terms, we reduce the expression to its simplest form, making calculations easier and more intuitive.

Algebraic Manipulation

Algebraic manipulation involves various techniques to transform and simplify expressions or equations.
Some common techniques include using the distributive property, combining like terms, and factoring.
In the given problem, we employ these techniques to simplify the expression step-by-step.
We start by distributing 18 across the terms inside the parentheses.
Then, we simplify these terms, followed by substituting back the simplified terms into the original expression.
Lastly, we combine like terms to get the final simplified expression, \( 15h - 13 \).
Mastering these algebraic manipulation skills is essential for solving more complex algebraic problems efficiently.
Practice and familiarity with these techniques will make algebra seem less daunting and more approachable.