Problem 81 - Elementary Algebra

Question

Simplify. $$ -7(10 x-7 y)-6(8 x+4 y) $$

Step-by-Step Solution

Answer

The simplified expression is \(-118x + 25y\).

Step 1: Distribute the Terms

Apply the distributive property to the expression \[-7(10x - 7y) - 6(8x + 4y)\].This means multiplying each term inside the parentheses by the factor outside them.\[-7(10x) + 7(7y) = -70x + 49y\] and \[-6(8x) - 6(4y) = -48x - 24y\].After distributing, the expression becomes:\[-70x + 49y - 48x - 24y\].

Step 2: Combine Like Terms

Combine the like terms from the expanded expression.- Combine the x terms: \[-70x - 48x = -118x\].- Combine the y terms: \[49y - 24y = 25y\].The expression simplifies to:\[-118x + 25y\].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify complex mathematical expressions. It states that the product of a number and a sum is the same as the sum of the products of the number and each addend separately. For example, when you encounter an expression like \(a(b + c)\), applying the distributive property, you will multiply \(a\) by both \(b\) and \(c\), resulting in \(ab + ac\). This property is especially useful when working with algebraic expressions that involve parentheses.

In the exercise, you apply this property to evaluate \(-7(10x - 7y)\) and \(-6(8x + 4y)\). By carefully distributing \(-7\) across \(10x\) and \(-7y\), first you get \(-70x + 49y\). Repeat the process for \(-6(8x + 4y)\), and you obtain \(-48x - 24y\).

This step results in breaking down the expression into separate terms, so that further simplification becomes manageable.

Combine Like Terms

Once you have distributed and expanded the terms, the next step is to identify and group the 'like terms.' Like terms refer to terms that have the same variables raised to the same powers. For instance, \(3x\) and \(5x\) are like terms because they both have \(x\) as the variable.

In our exercise, you are tasked to combine the terms that share the same variable components. Begin with the terms containing \(x\): \(-70x\) and \(-48x\). Add these together to get \(-118x\). The same method applies to the \(y\) terms: \(49y\) and \(-24y\) that combine to form \(25y\).

Combining like terms effectively reduces the complexity of the expression and brings you a step closer to obtaining the simplest form.

Simplification

Simplification is the process of transforming a complex expression into its simplest form. This process often involves several steps, including distributing terms and combining like terms.

After you've distributed and combined like terms, you're left with a simplified form of the expression. In our specific exercise, simplifying \(-70x + 49y - 48x - 24y\) results in \(-118x + 25y\). This showcases the power of simplification, which makes expressions more comprehensible and easier to work with.

A simplified expression is advantageous as it reduces the potential for errors in further calculations and helps in understanding the core components of the expression. The goal is to arrive at an expression where no further operations can combine like terms or reduce coefficients.