Problem 29 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 2+9 x=-7 $$

Step-by-Step Solution

Answer

The solution is \(x = -1\).

Step 1: Isolate the term with the variable

To solve the equation for \(x\), we first want to isolate the term that includes \(x\). Begin by subtracting \(2\) from both sides of the equation: \[2 + 9x - 2 = -7 - 2\]This simplifies to \[9x = -9\]

Step 2: Solve for the variable

Now, we need to solve for \(x\) by dividing both sides of the equation by \(9\):\[\frac{9x}{9} = \frac{-9}{9}\]This results in \[x = -1\]

Step 3: Verify the solution

Substitute \(x = -1\) back into the original equation to check if the solution is correct:\[2 + 9(-1) = -7\]Calculate the left side:\[2 - 9 = -7\]Since both sides are equal, \(x = -1\) is indeed a valid solution.

Key Concepts

Isolate the Variable

When tackling linear equations, the key is to 'isolate the variable'. This means getting the variable, often denoted as \(x\), by itself on one side of the equation. In our example, the equation is \(2 + 9x = -7\). To isolate the term with \(x\), we start by reversing any addition or subtraction that affects \(x\). You can do this by performing the opposite operation.
In this case, we need to subtract \(2\) from both sides of the equation. This step clears the constant on the same side as \(x\), simplifying our equation to \(9x = -9\).
The goal here is to have \(9x\), or the term with \(x\), alone, so you can easily solve for \(x\) in the next step. It's a simple rule-playing game of mathematics where balance is the aim.

Solving Equations

Solving equations is like finding a mystery number that makes the equation true. Once you've isolated the variable term, you are closer to solving the equation. For the equation \(9x = -9\), the next step is to find the value of \(x\). This is done through division.
Since \(9\) is being multiplied by \(x\), you want to do the opposite operation. Divide both sides by \(9\) to solve for \(x\). This will give you:

• \(\frac{9x}{9} = \frac{-9}{9}\)

This simplifies to \(x = -1\).
Now, \(x\) stands alone, and you've solved the equation by identifying the value that makes the equation true. Remember, solving equations involves clear logical steps to ensure every move maintains the equality.

Verifying Solutions

After solving the equation, verifying solutions is crucial to ensure correctness. It means plugging the solution back into the original equation to check if it holds true. For example, in the equation \(2 + 9x = -7\), once we found \(x = -1\), we insert it back to see:

Calculate \(2 + 9(-1)\) which should equal \(-7\). Here's the breakdown:

• \(2 + (-9) = -7\).

This confirms that both sides of the equation are equal, meaning \(x = -1\) is indeed correct.
This step might seem trivial, but it's paramount to verify our solution, ensuring no errors were made during the solve.