Problem 10 - Algebra 1: Concepts and Skills

Question

Solve the equation if possible. Determine whether the equation has one solution, no solution, or is an identity. $$ 5(x-5)=5 x+24 $$

Step-by-Step Solution

Answer

The given equation \(5(x-5) = 5x +24\), after simplification, gives a false statement. It means the equation has no solution.

Step 1: Distribute the coefficient on left side

To simplify the equation, the first step is to distribute the 5 on the left side of the equation. This is done by multiplying 5 to both \(x\) and \(-5\). This gives the following: \(5x - 25 = 5x + 24\).

Step 2: Rearranging equation

Further simplify the equation by re-arranging it. This is done by subtracting \(5x\) from both sides of the equation. This will give the following: \(-25 = 24\).

Step 3: Analysis of the result

Looking at the final simplified statement \(-25 = 24\), it is clear that this is a false statement. Hence, the original equation has no solution.

Key Concepts

Distributive Property

When dealing with algebraic equations, the distributive property is a useful tool that allows you to break down expressions inside parentheses. It's best understood with one simple rule: you multiply the term outside the parenthesis by each term inside the parenthesis. This is written as: \( a(b + c) = ab + ac \). This breaks down a complex expression into smaller, manageable pieces.

In the original exercise, the distributive property is used on the left side of the equation \( 5(x - 5) = 5x + 24 \). Here, 5 multiplies both \( x \) and \( -5 \), resulting in the expression \( 5x - 25 \).

• Multiply 5 with \( x \): \( 5 \times x = 5x \)
• Multiply 5 with \( -5 \): \( 5 \times -5 = -25 \)

By distributing, you simplify the expression inside the parentheses and make the equation easier to solve. This principle is essential whenever an equation includes parentheses, as it helps you clear away any nesting, allowing for straightforward manipulation of terms.

Solving Linear Equations

Solving linear equations involves isolating the variable to find its value. In our example, after distributing the terms in the beginning, we face the equation \( 5x - 25 = 5x + 24 \).

To solve such an equation, we need to get all terms involving the variable on one side and constant terms on the other. Here are the steps:

• Subtract \( 5x \) from both sides: This helps to see what part of the equation has the variables. Doing so gives \( -25 = 24 \).
• Analyze the resulting equation: If the variable cancels out, you end up with a statement about constants as in this case.

In linear equations, this process typically reveals the solution or allows you to determine if there is one. A linear equation can have one solution, no solution, or infinitely many solutions (identity). This analysis comes in the next section.

Identities and Solutions

Once we've re-arranged and simplified the equation, it's crucial to determine its solution status. Solutions can be categorized as follows:

• **One solution**: Occurs when the equation simplifies to \( x = a \), where \( a \) is a real number. This means the equation is solvable and has a distinct answer.
• **No solution**: Occurs when the equation simplifies to a false statement, such as \( -25 = 24 \). In this situation, no value of \( x \) will satisfy the original equation. It's like saying something impossible, setting no ground for any solution.
• **Identity (Infinitely many solutions)**: Occurs when the equation simplifies to a true statement about numbers, such as \( 0 = 0 \). This signifies that every value of \( x \) is a solution, as the truth of the equation does not depend on any specific value of \( x \).

In our exercise, upon simplifying, the resulting false statement \( -25 = 24 \) indicates that there is no solution. Recognizing these solution types not only helps you solve equations but also gives insight into their structure and the relationships between variables involved.