Problem 67 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ -3 x=9 $$

Step-by-Step Solution

Answer

The solution is \(x = -3\).

Step 1: Identify the Operation

The equation given is \[-3x = 9\]To isolate \(x\), we need to identify the operation affecting \(x\). Here, \(x\) is being multiplied by \(-3\).

Step 2: Perform the Inverse Operation

To solve for \(x\), we need to undo the multiplication by \(-3\) by performing the inverse operation. The inverse of multiplication is division. Divide both sides of the equation by \(-3\):\[\frac{-3x}{-3} = \frac{9}{-3}\]

Step 3: Simplify Both Sides

Now simplify both sides of the equation. On the left side, \(-3x / -3\) simplifies to \(x\).On the right side, \(9 / -3\) simplifies to \(-3\).So, we have \[x = -3\]

Step 4: Verify the Solution

To ensure our solution is correct, substitute \(x = -3\) back into the original equation:\[-3(-3) = 9\]Simplify to get \[9 = 9\]Since both sides are equal, our solution is verified.

Key Concepts

Inverse Operations

In mathematics, an important skill is understanding inverse operations. Inverse operations are pairs of operations that "undo" each other. This means that if you apply one operation and then its inverse to a number, you will get back to the original number.

For example:

• Addition and Subtraction - If you add 5 and then subtract 5, you get back to where you started.
• Multiplication and Division - If you multiply by 4 and then divide by 4, you also return to the original number.

In the context of solving equations, using inverse operations is a fundamental technique. They allow us to isolate the variable and find its value. When you have an equation involving multiplication, like \(-3x = 9\), you know that multiplication is the operation in effect. Thus, to solve it, you should apply the inverse, which is division, to both sides. This helps in systematically "undoing" the multiplication and achieving the solution.

Multiplication and Division

The operations of multiplication and division are interconnected. To illustrate, if you have an equation such as \(-3x = 9\), multiplication is clearly seen between \(-3\) and \(x\). Hence, solving such an equation involves performing division, the inverse operation, to unravel it.

Here's how it works step-by-step:

• Identify that \(x\) is multiplied by \(-3\).
• Use division, the inverse of multiplication, to isolate \(x\).
• Divide both sides of the equation by \(-3\): \(\frac{-3x}{-3} = \frac{9}{-3}\).

Once you perform these steps, you simplify the left side, eliminating the multiplication: \(-3x / -3\) becomes \(x\). On the right side, division resolves the arithmetic: \(9 / -3 = -3\). Hence, \(x = -3\), giving us a clean and clear result.

Verifying Solutions

Verifying solutions is a crucial step in solving equations. It ensures that the solution is accurate and adheres to the original equation's terms. To verify the solution of an equation like \(-3x = 9\), where you have found \(x = -3\), follow these steps:

Substitute the solution back into the original equation:

• Replace \(x\) with -3: \(-3(-3) = 9\).
• Simplify to check equality: \(9 = 9\).

If both sides of the equation are equal, it confirms that the solution is correct. Verification is essential as it acts as a double-check, assuring you that no arithmetic errors were made during the process. By routinely verifying, you build a stronger foundation in solving equations and boost confidence in your mathematical skills.