Problem 69 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ -5 x=-5 $$

Step-by-Step Solution

Answer

The solution is \(x = 1\).

Step 1: Identify the Equation Type

The given equation is a linear equation, which can be recognized by the presence of the variable raised to the first power.

Step 2: Isolate the Variable

To solve the equation \(-5x = -5\), we need to isolate \(x\). This can be done by dividing both sides of the equation by -5.

Step 3: Perform the Division

Divide both sides by -5:\[\frac{-5x}{-5} = \frac{-5}{-5}\]which simplifies to \(x = 1\).

Step 4: Check the Solution

Substitute \(x = 1\) back into the original equation to verify:\(-5(1) = -5\),which is true, confirming that \(x = 1\) is indeed the correct solution.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra. These equations represent a straight line when graphed and typically take the form \(ax + b = c\). In this case, the equation \(-5x = -5\) is a simple linear equation because the variable \(x\) is raised to the power of one. Linear equations like these have variables that appear only in the first degree.

Some common examples of linear equations include \(y = 2x + 3\) or \(3x - 7 = 5\). Solving these equations involves finding a specific value for the variable that makes the equation true. These equations serve as the building blocks for more complex algebraic concepts, making mastering their solution techniques essential for any math student.

Isolating the Variable

Isolating the variable is an essential step in solving linear equations. This process involves manipulating the equation to get the variable by itself on one side of the equation. For the equation \(-5x = -5\), we want to find the value of \(x\) that makes the equation true.

Here are the steps to isolate the variable in the given problem:

• Recognize that \(x\) is being multiplied by \(-5\). To undo this, perform the opposite operation. This means dividing both sides of the equation by \(-5\).
• Apply the division to both sides, like this: \(\frac{-5x}{-5} = \frac{-5}{-5}\).
• Simplify the equation, resulting in \(x = 1\).

Isolating the variable often involves reversing operations: if the variable is added, subtract; if it's multiplied, divide. This skill is crucial to solving any equation accurately.

Verifying Solutions

Verifying solutions is the final step in solving linear equations and ensures that the solution is correct. After isolating the variable and finding a potential value, it's important to substitute it back into the original equation. This step helps confirm the accuracy of your work.

Let's verify the solution \(x = 1\) for the equation \(-5x = -5\):

• Substitute the value of \(x\) back into the equation: \(-5(1) = -5\).
• Calculate the left side: \(-5 \times 1 = -5\), which matches the right side of the equation.

Because both sides of the equation match when \(x = 1\), the solution is correct. Verifying solutions provides confidence that you've performed all steps correctly and arrived at the proper answer.