Problem 62 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ -4(1-x)+7 $$

Step-by-Step Solution

Answer

The simplified expression is \(4x + 3\).

Step 1: Distribute negative four

Distribute the -4 to each term within the brackets. After the distribution, the expression changes to: -4*1 + -4*(-x). Therefore, the expression becomes: -4 + 4x.

Step 2: Combine like terms

Combine the constant term -4 and 7. The expression is simplified to: 4x + 3.

Key Concepts

Distributive Property

The distributive property is a fundamental principle used in algebra that allows you to break down expressions. It states that if you have something like \( a(b+c) \), it's the same as \( ab + ac \).
When applying the distributive property, you multiply the term outside the parentheses by each term inside. Let's see how it worked in our original exercise.

• We started with \(-4(1-x)\).
• Using the distributive property, we multiply \(-4\) by \(1\), which gives us \(-4\).
• Next, we take \(-4\) times \(-x\), leading to \(+4x\).

These multiplying steps convert the original expression into \(-4 + 4x\).
Distributing can help break down expressions into simpler components, making them easier to solve.

Combining Like Terms

Like terms are terms in an expression that have the same variables raised to the same power. These terms can be combined, or added together, because they are like apples to apples. In our exercise, after using the distributive property, we ended up with the expression \(-4 + 4x\).

• Here, \(-4\) and \(7\) are considered like terms because they are both constant numbers without any variables.
• By adding them together, we get \(3\). Thus, the entire expression becomes \(4x + 3\).

Combining like terms simplifies the expression further. It ensures that all similar parts of the expression are condensed into one, giving a cleaner and more readable expression.

Linear Expressions

A linear expression is an algebraic statement in which each term is either a constant or the product of a constant and a single variable raised to the first power. In our case, after simplification, the expression \(4x + 3\) is a perfect example of a linear expression.

• The term \(4x\) includes the variable \(x\) raised to the first power, making it linear.
• The \(+ 3\) is a constant, which is also part of many linear expressions.

Linear expressions are called "linear" because they graph as a straight line, which is simple yet powerful. They form the basis for more advanced algebraic concepts you'll encounter later. Understanding these linear terms is crucial in recognizing relationships between variables and constants in algebra.