Problem 57 - Algebra 1

Question

Simplify the variable expression. $$4(3 x+5 y)+3(-4 x+2 y)$$

Step-by-Step Solution

Answer

The simplified expression is \(26y\).

Step 1: Apply the Distributive Property

We have an expression \(4(3 x+5 y)+3(-4 x+2 y)\). By applying the distributive property (i.e. multiplying the outside number with each term inside the parentheses), this expression becomes \(12x + 20y -12x + 6y\).

Step 2: Combine Like Terms

Next, we combine the like terms in the equation. Like terms are terms whose variables (and their exponents such as the 2 in \(x^2\)) are the same. In this expression, the like terms are the terms with \(x\) and the terms with \(y\). We combine these to form \(0x + 26y\).

Step 3: Simplify to Obtain Final Expression

A variable term multiplied by zero disappears from the equation, so our final expression is \(26y\).

Key Concepts

Distributive Property

The Distributive Property is a fundamental aspect of algebra that allows us to simplify expressions efficiently. Imagine you have the expression \(4(3x + 5y) + 3(-4x + 2y)\). Here, we apply the distributive property by distributing, which means multiplying each term inside the parentheses by the factor outside. This operation transforms the expression into \(12x + 20y - 12x + 6y\).

Breaking it down:

• Multiply the \(4\) by each term within the first set of parentheses: \(4 \times 3x = 12x\) and \(4 \times 5y = 20y\).
• Next, do the same with the \(3\) for the second set of parentheses: \(3 \times -4x = -12x\) and \(3 \times 2y = 6y\).

Utilizing this property helps in reorganizing the expression, making it a simple matter of basic arithmetic to carry out further operations.

Combining Like Terms

Combining like terms is the process of simplifying an expression by merging terms that have identical variables and powers. In the expression \(12x + 20y - 12x + 6y\), we need to collect terms that look alike—in this case, those with \(x\) and those with \(y\).

• For the \(x\)-terms: \(12x - 12x\) combine to give \(0x\), which essentially eliminates the \(x\) component.
• For the \(y\)-terms: \(20y + 6y\) combine to produce \(26y\).

Combining like terms simplifies expressions by reducing them to fewer terms, making them easier to manipulate in further calculations.

Consider like terms as combining identical features in a set—the same way you might group similar fruits together in a basket.

Simplification

Simplification in algebra is about reducing expressions to a more manageable and understandable form. Once we have an expression where terms have been distributed and like terms combined, our job is to ensure it is as concise as possible.

• In the expression \(0x + 26y\), the term \(0x\) disappears because multiplying any number by zero results in zero.
• Thus, we are left with \(26y\), which is the simplest form of the original expression.

This process helps highlight essential parts of the expression without losing any mathematical substance.

Simplifying expressions lays a foundation to analyze, solve, and graph equations, equipping us with a streamlined result to work with in various mathematical problems.