Problem 124 - College Algebra

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 'The equation -7x = x has no solution.' is false. The true statement should be 'The equation -7x = x has a solution, which is x = 0.'

Step 1: Write down the original equation

The original equation given is -7x = x.

Step 2: Rearrange the equation

Rearrange the equation to determine if a solution is present. This can be done by adding 7x to both sides of the equation to get 0 = x + 7x, which simplifies to 0 = 8x.

Step 3: Solve for x

Following the principle of algebra, the equation 0 = 8x can be divided by 8 to isolate the variable x. After dividing, we find that x = 0.

Step 4: Evaluate the original equation with the found solution

Substitute x = 0 back into the original equation to see if it holds. The original equation -7x = x becomes -7*0 = 0 which simplifies to 0 = 0, which is a true statement.

Step 5: Determine the truthfulness of the original statement

Since a solution (x = 0) to the equation -7x = x does exist, the statement 'The equation -7x = x has no solution' is false. The statement can be made true by saying 'The equation -7x = x has a solution'.

Key Concepts

True or False Statements

When we encounter a mathematical statement, such as an equation, it is often important to determine whether it is true or false. This involves examining the given proposition and checking it against established mathematical principles. In our problem, the statement was that the equation \(-7x = x\) has no solution.
Here are steps we follow to verify statements of this kind:

• Firstly, clearly define what the statement claims. In this case, whether there are any values of \(x\) that make the equation true.


• Next, try to solve the equation to verify if any such value exists.


• If you find a value that satisfies the equation, the statement is false. If no value satisfies the equation, it's true.

In the exercise, solving the equation reveals that \(x = 0\) is a valid solution, contradicting the original statement about the lack of solutions. Thus, the statement "The equation \(-7x = x\) has no solution" is false.

Solving Linear Equations

Linear equations are a type of equation that involve variables raised only to the first power. Solving linear equations entails finding the value of the variable that makes the equation true. For example, we started with the equation \(-7x = x\).
Here are some steps to solve linear equations effectively:

• Isolate the variable on one side by performing inverse operations like addition, subtraction, multiplication, or division.


• Add 7x to both sides of the equation to eliminate \(-7x\) on one side, resulting in \(0 = 8x\).


• Divide both sides by the coefficient of the variable, 8 in this case, to solve for \(x\).

Applying these steps to our problem led to the solution \(x = 0\). Understanding these fundamental steps is crucial because linear equations appear frequently in algebra and other branches of mathematics.

Algebraic Manipulation

Algebraic manipulation involves performing operations to rearrange and simplify expressions or equations. It is an essential skill in solving equations and verifying statements. In the problem, we manipulated the original equation \(-7x = x\) to find a solution.
Here's how algebraic manipulation is applied:

• Start with the original form. Identify like terms to combine or rearrange.


• In the equation \(-7x = x\), adding \(7x\) to both sides combined the variables, leading to \(0 = 8x\).


• By dividing both sides by 8, the equation simplifies, isolating \(x\).

Algebraic manipulation not only makes the problem more manageable but also guides you to the correct solution. Mastery of these techniques ensures accuracy in equations like these and many other mathematical tasks.