Problem 47 - Algebra A Combined Function

Question

Solve. See Examples 1 through 7 $$ 5 x-5=2(x+1)+3 x-7 $$

Step-by-Step Solution

Answer

The equation has infinitely many solutions.

Step 1: Distribute and Simplify

First, distribute the 2 across the terms inside the parentheses on the right-hand side of the equation: \[ 2(x + 1) = 2x + 2 \]So the equation becomes:\[ 5x - 5 = 2x + 2 + 3x - 7 \].Next, simplify the right-hand side by combining like terms:\[ 2x + 3x = 5x \] and \[ 2 - 7 = -5 \]. The equation now is:\[ 5x - 5 = 5x - 5 \].

Step 2: Analyze the Equation

Examine the simplified equation \( 5x - 5 = 5x - 5 \). Since both sides are identical, it is evident that every real number is a solution to this equation. There are no steps needed to isolate \( x \), as the equality holds regardless of the value of \( x \).

Step 3: Conclude with Infinite Solutions

Since the equation \( 5x - 5 = 5x - 5 \) holds true for all potential values of \( x \), we conclude that there are infinitely many solutions.

Key Concepts

Equation Simplification

The process of simplifying equations is a fundamental skill in algebra. It involves transforming a given equation into its simplest form, making it easier to understand and solve. Simplification typically involves combining like terms and handling operations such as addition, subtraction, multiplication, and division.

In the example problem, after applying the distribution property, we have terms like \(2x + 3x\) and \(2 - 7\) that need to be simplified. To simplify these:

• We combine like terms, such as \(2x\) and \(3x\), resulting in \(5x\).
• We also simplify the numbers \(2\) and \(-7\) to get \(-5\).

This leads us to a simplified equation where both sides are \(5x - 5\), which reveals much about the solution set.

The goal is to reach a step where the equation is concise and as straightforward as possible, often preparing it for further analysis in solving for variables.

Infinite Solutions

In the journey of solving linear equations, sometimes we encounter equations with infinite solutions. This happens when an equation is true for all values of the variable involved.

In our simplified equation, \(5x - 5 = 5x - 5\), you can see that both sides of the equation are identical. No matter what value \(x\) takes, substituting it into the equation will hold the equality true.

• This type of equation hints that there isn't just one solution, but a multitude of them—literally any real number can be a solution.
• This outcome is different from the more common single solution case, where there is a specific number that satisfies the equation.

Recognizing infinite solutions is crucial, as it can help in understanding the behavior and characteristics of equations in various contexts.

Whenever you see both sides of an equation are the same after simplification, you can directly conclude that there are infinite solutions without further calculation.

Distribution Property in Algebra

The distribution property, also known as the distributive property, is a key tool in algebra used to simplify expressions. It states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately.

The general rule is: \(a(b + c) = ab + ac\). In our exercise, this is applied when dealing with \(2(x + 1)\), which becomes \(2x + 2\) after distribution:

• This property allows us to "spread" the multiplication over each term within the parentheses.
• Making sure to apply this property correctly is vital as it ensures the equation is transformed accurately.

Without correctly using the distribution property, one might end up with erroneous results, leading to incorrect solutions.

Understanding and applying the distribution property helps pave the way for further simplification and accurate equation solving, especially in complex algebraic expressions.