Problem 17 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ -6 a=48 $$

Step-by-Step Solution

Answer

The solution is \(a = -8\).

Step 1: Identify the equation

The given equation is \(-6a = 48\). Our goal is to find the value of \(a\).

Step 2: Isolate the Variable

To solve for \(a\), we need to isolate it on one side of the equation. Divide both sides of the equation by \(-6\) to get:\[a = \frac{48}{-6}\]

Step 3: Simplify the division

Simplify \(\frac{48}{-6}\) by performing the division:\[a = -8\]

Step 4: Verify the solution

Substitute \(a = -8\) back into the original equation to check if it holds true:\[-6(-8) = 48\]Simplifying the left side gives \(48 = 48\), which confirms that the solution is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in mathematics. It involves finding the value of a variable that makes an equation true. This process is akin to solving a puzzle, where each piece comes together to reveal the final picture. When tackling an equation, such as the one given, \(-6a = 48\), the aim is to determine the value of \(a\) that satisfies the equation.

To begin solving, it's important to understand both sides of the equation. On one side, you have \(-6a\), indicating that \(-6\) is multiplied by our unknown variable \(a\).

• The left side represents a mathematical operation involving the variable.
• The right side, 48, signifies the result of this operation.

Recognize the balance between both sides of the equation as a scale, where changes need to be equally applied. This makes sure the equation remains valid.

Solving equations becomes more intuitive with practice, and each equation you solve lays the foundation for more complex concepts.

Isolation of Variable

The isolation of variables is a crucial step in solving equations. It involves manipulating the equation to have the unknown variable by itself on one side of the equation.

Looking at our specific problem, \(-6a = 48\), our goal is to solve for \(a\). We need to "free" \(a\) from being multiplied by \(-6\).

To achieve this, we perform the operation that "undoes" multiplication: division. Divide both sides of the equation by \(-6\):

• Divide the left side: \(-6a \div -6 = a\)
• Divide the right side: \(48 \div -6 = -8\)

This operation results in the equation \(a = -8\). Now, \(a\) is isolated, meaning it stands alone, which fulfills our goal.

By isolating the variable, we have derived the solution, revealing the value that satisfies the original equation.

Verification of Solutions

Verification is the process of checking that the solution derived is correct. This step assures that no errors occurred during the solving process and that the answer makes sense in the context of the problem.

For our equation, \(-6a = 48\), we found \(a = -8\) by isolating the variable. To verify it, substitute \(a = -8\) back into the original equation:

• Left side: \(-6(-8)\)
• Right side: 48

Calculate the left side: \(-6 \times -8 = 48\).

Both sides match, confirming that the solution is accurate. Thus, the answer \(a = -8\) is indeed correct.

Verification is crucial in mathematics as it boosts confidence in the results, ensuring that all calculations leading to the solution were performed accurately.