Problem 124 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change \((s)\) to produce a true statement. The equation \(-7 x=x\) has no solution.

Step-by-Step Solution

Answer

The statement is false. The equation -7x = x has a solution which is x = 0.

Step 1: Solve the Equation

Given the equation, \(-7x = x\). First, bring \(x\) from the right side of the equation to left by adding \(-x\) on both sides to get \(-7x - x = 0\). Combine similar terms to form a new equation, \(-8x = 0\).

Step 2: Determine x value

To determine \(x\), divide the entire equation by \(-8\) to isolate \(x\). The equation then becomes \(x = 0 / -8\). Solving this provides \(x = 0\).

Step 3: Comment on the Statement

The statement claims that the equation \(-7x = x\) has no solution. However from the solution derived in step 2, \(x = 0\) is a valid solution for the given equation. Therefore, the statement is false.

Key Concepts

Understanding the Properties of Equality

Whenever you solve linear equations, it's important to apply the properties of equality. These properties provide the logic needed to manipulate equations and are founded on the principle that if two things are equal, performing the same operation on both sides will keep the equation balanced.
You can think of it like a seesaw. If both sides are balanced with equal weight, adding or removing the same weight from each side keeps it balanced. In equations, this means:

• Addition Property: If you add the same number to both sides, the equality is maintained. For instance, if you have the equation \(a = b\), then \(a + c = b + c\).
• Subtraction Property: If you subtract the same number from both sides, the equality remains true. If \(a = b\), then \(a - c = b - c\).
• Multiplication Property: Multiplying both sides by the same non-zero number keeps the equation true. So, \(a = b\) leads to \(a \, \cdot \, c = b \, \cdot \, c\).
• Division Property: Similarly, dividing both sides by the same non-zero number will not affect the equation's truth. From \(a = b\), you can get \(a / c = b / c\).

These properties are key tools when solving equations and are used repeatedly in the process to simplify and resolve the equation.

The Process of Solving Equations

Solving linear equations involves finding the value of the unknown variable that makes the equation true. It's like a detective investigation where you search for the missing piece. To solve equations effectively, follow these steps:

• Simplify: Remove any parentheses by expanding and combine like terms to make the equation easier to work with.
• Isolate the Variable: Use the properties of equality to move terms containing the variable to one side and constants to the other. This often involves adding, subtracting, multiplying, or dividing both sides of the equation.
• Perform Operations: Once the variable is isolated, simplify the final expression. If an operation involves division, make sure the divisor is not zero, as division by zero is undefined.

In our exercise, starting with the equation \(-7x = x\), we used properties of equality to reorganize and simplify the equation to \(-8x = 0\). This led to isolating \(x\) by dividing both sides by \(-8\), providing us the solution \(x = 0\).

Understanding Zero as a Solution

When solving linear equations, a special case arises when the solution is zero. Many students mistakenly believe zero means no solution, but it's actually a valid number and perfectly fine as a solution.
Let's explore what it means for zero to be a solution in a mathematical equation. If you uncover that \(x = 0\) is a solution, it simply means substituting \(0\) for \(x\) satisfies the equality condition of the equation.

• For example, in the equation \(-7x = x\), when we substitute 0 for \(x\), the equation becomes \(-7 \times 0 = 0\), which is a true statement.
• Finding that zero is the solution indicates the equation has one solution, as opposed to none or infinitely many.

Remember, just because zero looks insignificant, it doesn't diminish its value or correctness as an answer in mathematical contexts.