Problem 88 - Intermediate Algebra

Question

Solve each equation. $$ -2(x+5)=x+30-2 x $$

Step-by-Step Solution

Answer

The value of x is -40.

Step 1: Simplify the equation

First, simplify both sides of the equation if needed. On the left side, distribute -2ackslash into x + 5ackslash:, which means multiply -2ackslash by both xackslash and 5ackslash.This gives: \[-2(x + 5) = -2x - 10\] Now rewrite the equation using this simplified form:\[-2x - 10 = x + 30 - 2x\].

Step 2: Combine like terms

Look at the right side of the equation. You have x - 2xackslash, which simplifies to -xackslash. The equation becomes:\[-2x - 10 = -x + 30\].

Step 3: Isolate variable terms

Add 2xackslash to both sides of the equation to move all xackslash terms to one side:\[-2x - 10 + 2x = -x + 30 + 2x\] which simplifies to:\[-10 = x + 30\].

Step 4: Solve for the variable

Subtract 30ackslash from both sides to solve for xackslash:\[-10 - 30 = x + 30 - 30\], which simplifies to:\[-40 = x\].

Step 5: Final Result

We have solved the equation for xackslash and found that x = -40ackslash.

Key Concepts

Distributive Property

The distributive property is a powerful tool in algebra that helps simplify expressions. It allows you to remove parentheses by distributing one term across the others inside the parentheses.
This means you multiply the term outside the parentheses by each term inside. For example, given an expression like \(-2(x + 5)\), the \(-2\) is distributed to both \(x\) and \(5\).

• The expression \(-2(x + 5)\) becomes \(-2 \cdot x + (-2) \cdot 5\).
• After multiplication, this results in \(-2x - 10\).

Using the distributive property helps simplify more complex expressions and bring them into a form where other algebraic methods can be applied. Once simplified, the equation becomes easier to solve.

Combining Like Terms

Combining like terms is a method used to simplify equations by merging terms that have the same variable raised to the same power.
This step is essential in clarifying the expression during the solving process.

• Consider the expression \(-2x - 10 = x + 30 - 2x\).
• The terms \(x - 2x\) on the right are like terms since they both contain the variable \(x\).
• Combine them to get \(-x\).

This simplification changes the equation to \(-2x - 10 = -x + 30\).
By reducing the number of terms, the equation becomes easier to manage. It's an important step before isolating the variable.

Isolating Variables

Isolating the variable means rearranging the equation to get the unknown variable by itself on one side of the equation. This process involves using basic operations such as addition, subtraction, multiplication, or division.
In this equation, all \(x\) terms should be on one side. Here's how to do it:

• Add \(2x\) to both sides to move the \(x\) term from the left to the right.
• This transforms the equation from \(-2x - 10 = -x + 30\) to \(-10 = x + 30\).

The goal is to have the variable \(x\) on one side by itself so that further operations can provide its value.
Isolating a variable is a skill that becomes second nature with practice and is foundational in solving linear equations.

Simplifying Equations

Simplifying an equation is about reducing it to its most basic form until you can easily solve it for a particular variable.
This involves boiling down unnecessary terms and ensuring every step logically follows from the last.

• Once you've isolated \(x\) in the equation \(-10 = x + 30\), you simplify by removing the constant on the right side.
• Subtract 30 from each side to balance the equation: \(-10 - 30 = x + 30 - 30\).
• Simplifying gives \(-40 = x\).

This clear and concise form makes it easy to identify the solution.
Simplifying equations is not just an algebraic technique—it's a thought process that brings order and clarity in solving mathematical problems.