Problem 22 - Elementary Algebra

Question

Solve the equations. $$ 9 a+5=-22 $$

Step-by-Step Solution

Answer

Answer: -3

Step 1: Identify the equation

We are given the following linear equation to solve for a: $$ 9a + 5 = -22 $$

Step 2: Use inverse operations to isolate the variable "a"

To solve the equation for the variable a, we need to isolate it by using inverse operations. The inverse operation of addition is subtraction, and the inverse operation of multiplication is division.

Step 3: Subtract 5 from both sides of the equation

Since the equation has a constant term of 5 added to 9a, we will subtract 5 from both sides of the equation to cancel out the constant term on the left side. This will give us: $$ 9a + 5 - 5 = -22 - 5 $$ Simplifying, we have: $$ 9a = -27 $$

Step 4: Divide by 9 to solve for "a"

Now that we have 9a equals to a constant, we can divide both sides of the equation by 9 to isolate the variable a. We will get: $$ \frac{9a}{9} = \frac{-27}{9} $$ Simplifying, we have: $$ a = -3 $$

Step 5: Write the final solution

The value of "a" in the given equation is -3, and our final solution is: $$ a = -3 $$

Key Concepts

Inverse Operations

In mathematics, inverse operations are essential tools that help us solve equations efficiently. They are operations that reverse the effect of another operation. In the context of solving equations, inverse operations are used to "undo" operations to simplify the equation and solve for a variable.
For example:

• The inverse of addition is subtraction.
• The inverse of multiplication is division.

When you encounter an equation like \(9a + 5 = -22\), you're looking at an equation where operations are applied to the variable \(a\). To isolate \(a\), you must perform operations that will eliminate these effects. This is where inverse operations come into play.
By identifying the operations impacting the variable, you can apply their inverses to both sides of the equation, gradually simplifying it. This method is the cornerstone of problem-solving in linear equations and ensures that variables are isolated correctly.

Variable Isolation

Variable isolation is a critical step in solving any equation. Simply put, it involves rearranging the equation to have the variable stand alone on one side. The ultimate goal is to express the unknown variable in terms of a known quantity.
In the example \(9a + 5 = -22\), our target is to isolate \(a\). Initially, \(a\) is being multiplied by 9 and added to 5. To isolate it, we need to remove these surrounding operations.

• Start with the most outer operation. Here, it's the addition of 5, so we subtract 5 from both sides, simplifying the equation to \(9a = -27\).
• Now, \(a\) is still being multiplied by 9. The next step is to divide both sides by 9 to completely isolate \(a\), giving us \(a = -3\).

By following these steps, we effectively isolate the variable, making it possible to read off its value from the equation.

Solving Equations

Solving equations is a process of finding the value of the variable that makes the equation true. It involves several steps guided by logic and mathematical rules.
Here’s a general approach to solving linear equations:

• Identify: Recognize the type of equation and the operations affecting the variable.
• Apply Inverse Operations: Use these operations to counteract any additions or multiplications impacting the variable.
• Isolate the Variable: Rearrange the equation to have the variable by itself on one side, thereby finding its value.
• Simplify: Perform any necessary simplifications to reach the simplest form of the solution.

In the equation \(9a + 5 = -22\), solving it involved inverse operations and isolating \(a\). Our steps followed a clear path: subtracting 5, simplifying to \(9a = -27\), and then dividing by 9 to finally find \(a = -3\). Each step must be carried out carefully, ensuring that the equations remain balanced and each operation is correctly reversed.