Problem 34 - Algebra A Combined Function

Question

Solve each equation. See Examples 6 and \(7 .\) $$ 2(x-5)=2 x+10 $$

Step-by-Step Solution

Answer

No solution; the equation results in a contradiction.

Step 1: Distribute the Left Side

Distribute the 2 on the left side of the equation. Starting with: \[ 2(x-5) = 2x + 10 \] Distribute the 2:\[ 2 \times x - 2 \times 5 = 2x + 10 \]This simplifies to: \[ 2x - 10 = 2x + 10 \]

Step 2: Eliminate Like Terms

Subtract \(2x\) from both sides to eliminate the \(2x\) terms:\[ 2x - 10 - 2x = 2x + 10 - 2x \]This simplifies to:\[ -10 = 10 \]

Step 3: Analyze the Result

The equation \(-10 = 10\) is incorrect, indicating a contradiction. This means there is no solution, as this is not interpretable as a valid equation.

Key Concepts

Equation Solving

Solving equations is a core part of algebra. Equations often involve finding the value of a variable that makes the expression true. In algebra, we use a step-by-step process to simplify and solve these equations.

• The goal is to isolate the variable on one side of the equation. This means you will perform operations like addition, subtraction, multiplication, or division on both sides to keep the equation balanced.
• It's important to perform the same operation on both sides of the equation. This keeps the equation equal and valid throughout the solution process.

In our exercise, we are attempting to solve the equation \(2(x-5)=2x+10\). The process will reveal if the equation has a solution or not. Sometimes, the solution may result in a scenario where no values satisfy the equation, as you'll see later.

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify equations. It allows you to multiply a single term by terms inside a parenthesis.

• The property works by distributing the multiplication over addition or subtraction inside the parenthesis. Mathematically, it is expressed as \(a(b + c) = ab + ac\).

In the given exercise, the distributive property helps us break down \(2(x-5)\):

• First, multiply the 2 by \(x\) and the 2 by \(-5\).
• This results in \(2x - 10\), transforming the equation to \(2x - 10 = 2x + 10\).

This property is crucial for simplifying equations, making them easier to solve or analyze further.

No Solution Scenarios

Sometimes in algebra, you might encounter equations that, after simplification, don't make logical sense. These are known as 'no solution scenarios.'

• A no solution scenario occurs when the two sides of an equation are always different numbers, meaning no possible value for the variable can make the equation true.

In our exercise, after distributing and simplifying, we ended up at an equation \(-10 = 10\). This equation is a contradiction; it cannot be true no matter what value is assigned to the variable. Hence, we conclude there is no solution.

• Recognizing such scenarios is key because it tells us that either the initial conditions were flawed, or in real-world context, that perhaps the problem constraints are impossible to satisfy.