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Chapter 2: Problem 12

For exercises 1-72, (a) solve. (b) check. $$ -x+9=14 $$

Short Answer

Expert verified
x = -5

Step by step solution

01

- Isolate the Variable

To solve the equation -x + 9 = 14, we need to isolate the variable x. First, subtract 9 from both sides of the equation to move the constant term to the other side: -x + 9 - 9 = 14 - 9 which simplifies to: -x = 5
02

- Solve for the Variable

Next, we solve for x by dividing both sides of the equation by -1: -x / -1 = 5 / -1 which simplifies to x = -5
03

- Check the Solution

To check the solution, substitute x = -5 back into the original equation to see if it holds true: -(-5) + 9 = 14 that simplifies to: 5 + 9 = 14 Since 14 = 14, the solution x = -5 is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate the Variable
To solve a linear equation, the first important step is to isolate the variable. In our exercise, the equation given is \(-x + 9 = 14\). Our goal is to get \(x\) by itself on one side of the equation. Start by moving the constant term to the other side. This is done by subtracting 9 from both sides of the equation:

\(-x + 9 - 9 = 14 - 9\), which simplifies to \(-x = 5\).

Isolating the variable is crucial because it simplifies the equation and helps us solve it.
Check the Solution
After solving for the variable, it's important to check the solution. This helps ensure that your answer is correct. Substitute the solution back into the original equation to verify it holds true.

For our solution, \(x = -5\), we substitute it back into the equation: \(-(-5) + 9 = 14\), which simplifies to \(5 + 9 = 14\). Since both sides of the equation are equal, our solution \(x = -5\) is verified.

Checking the solution is a good practice as it confirms the accuracy of your answer.
Simplify Equations
Simplifying equations is another key step in solving linear equations. You often need to perform basic arithmetic operations like addition, subtraction, multiplication, or division.

In our example, simplification occurs when we subtract 9 from both sides: \(-x + 9 - 9 = 14 - 9\) simplifies to \(-x = 5\).

Then, we divided both sides by \(-1\): \(-x / -1 = 5 / -1\) simplifies to \(x = -5\).

Always simplify each side of the equation step-by-step to make the problem more manageable.
Substitution Method
The substitution method involves replacing a variable with a known or calculated value to solve an equation.

In our check step, we used the substitution method to verify our solution: \(-(-5) + 9 = 14\). Here, \x = -5\ was substituted back into the original equation.

This step is important for verifying your solution and ensuring your calculations are correct.

Always remember to substitute your solution back into the original equation to see if it holds true.

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