Problem 19 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(7 x+8=1\)

Step-by-Step Solution

Answer

The solution is x = -1. It is neither an identity nor a contradiction.

Step 1 - Isolate the variable term

First, subtract 8 from both sides of the equation to isolate the term with the variable: 7x + 8 - 8 = 1 - 8 This simplifies to: 7x = -7

Step 2 - Solve for x

Next, divide both sides of the equation by 7 to solve for x: \( \frac{7x}{7} = \frac{-7}{7} \) This simplifies to: x = -1

Step 3 - Check the solution

Substitute x = -1 back into the original equation to verify the solution: 7(-1) + 8 = 1 Simplify: -7 + 8 = 1 Since 1 = 1 is a true statement, the solution x = -1 is correct.

Step 4 - Determine if it is an identity or a contradiction

Because the equation 7x + 8 = 1 has a single solution, x = -1, it is neither an identity nor a contradiction. An identity is an equation that is true for all values of the variable, and a contradiction is an equation that has no solution.

Key Concepts

Isolating the Variable

To solve a linear equation, the first step often involves isolating the variable. Here, the goal is to get the variable, in this case, x, by itself on one side of the equation. In the example 7x + 8 = 1, we start by removing the constant term (8) from the left side.

We achieve this by subtracting 8 from both sides:
7x + 8 - 8 = 1 - 8
This simplifies to: 7x = -7.

By isolating the variable, it becomes much easier to solve the equation. This step is essential because it transforms the equation into a simpler form, which is crucial for finding the solution.

Checking the Solution

Once you have found a potential solution for x, it's important to verify that it satisfies the original equation. This is done by substituting the solution back into the original equation.

In our example: we found x = -1. To check, substitute -1 for x in 7x + 8 = 1:
7(-1) + 8 = 1
Simplifying further: -7 + 8 = 1
Since both sides equal 1, our solution is verified.

Verifying the solution helps to ensure there are no mathematical errors and confirms that the found value for the variable is indeed correct.

Identity

An identity is an equation that is true for all values of the variable. This means that no matter what value you substitute for the variable, the equation remains true.

For example, the equation x - x = 0 is an identity because it holds true regardless of the value of x.

In the given problem, 7x + 8 = 1, we determined there is a single specific solution, x = -1. This equation is not an identity, since it only holds true for one specific value of x.

Contradiction

A contradiction is an equation that has no solution. It means there is no possible value for the variable that would make the equation true.

For example, the equation x + 2 = x + 3 is a contradiction, as no value for x will satisfy the equation.

In our problem, while solving 7x + 8 = 1, we found a valid solution: x = -1. Therefore, the equation is not a contradiction, since there is at least one solution that satisfies it.