Problem 31 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$9(7-a)$$

Step-by-Step Solution

Answer

The expression \(9(7-a)\) without parentheses using the distributive property is \(63 - 9a\).

Step 1: Identify the Terms

The given expression is \(9(7-a)\). Here, '9' is the coefficient and '(7-a)' is inside the parenthesis.

Step 2: Apply Distributive Property

Applying the distributive property, the expression can be rewritten by multiplying 9 with each term inside the parentheses. So, the equation becomes \(9 * 7 - 9 * a\).

Step 3: Simplify

Next, complete the multiplication operations. \(9*7 = 63\) and \(9*a\) will remain as is since a is a variable. The expression becomes \(63 - 9a\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators such as addition, subtraction, multiplication, and division. In the expression \(9(7-a)\), the parentheses indicate that the terms inside should be treated as one entity when applying operations. You can think of it like a cooking recipe where the ingredients inside the parenthesis are mixed together before being combined with other ingredients, in this case, the number '9'.

• An algebraic expression can have constants, which are fixed numbers like '7' in this case.
• It can also have variables, like 'a', which represent unknown values and can change.

The goal can often be to simplify the expression using mathematical properties, such as the distributive property, to make calculations easier or to solve for a variable.

Coefficients

Coefficients are the numbers in front of the variables in algebraic expressions. They tell us how many times the variable is being multiplied. In our example, the number '9' is a coefficient that is multiplying the entire expression inside the parenthesis \((7-a)\).

• When you apply the distributive property, you'll multiply the coefficient by every term inside the parenthesis.
• Here, '9' is multiplying both '7' and '-a'.

The final expression, \(63 - 9a\), shows '63' as the simplified constant, and '-9' as the coefficient of 'a'. This makes it clearer how the variable 'a' influences the expression's value based on its coefficient.

Simplifying Expressions

Simplifying expressions means making them easier to work with by performing all possible operations. After using the distributive property, expressions like \(9(7-a)\) become simpler and more straightforward, facilitating better understanding and further calculations.

To simplify, follow these steps:

• Apply relevant mathematical properties, such as the distributive property, to eliminate parentheses.
• Carry out multiplication and combine like terms if available.

In this exercise, we multiplied '9' with each term inside the parentheses to obtain \(63 - 9a\). By breaking the original expression into separate, simple pieces, we arrive at a clearer, more manageable form. Simplifying expressions makes algebra much easier to handle and makes it easier to see the relationships between terms.