Problem 8 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each solution. $$ -3 a-6=9 $$

Step-by-Step Solution

Answer

The solution is \(a = -5\).

Step 1: Add 6 to Both Sides

The equation given is \(-3a - 6 = 9\). To isolate the term with \(a\), first add 6 to both sides of the equation. By doing this, we get: \(-3a - 6 + 6 = 9 + 6\), which simplifies to \(-3a = 15\).

Step 2: Divide by -3

Now that we have \(-3a = 15\), divide both sides by -3 to solve for \(a\). This gives us: \(a = \frac{15}{-3}\), which simplifies to \(a = -5\). So, \(a = -5\) is our solution.

Step 3: Check the Solution

To ensure the solution is correct, substitute \(a = -5\) back into the original equation \(-3a - 6 = 9\). Substitute \(-5\) for \(a\): \(-3(-5) - 6\). This simplifies to \(15 - 6\), which equals 9. Since both sides of the equation are equal after substitution, the solution \(a = -5\) is verified to be correct.

Key Concepts

Checking Solutions

Once you've found a possible solution to an equation, it's crucial to confirm that it’s correct. This process is known as "checking solutions." It ensures you haven't made any errors during the solving steps.
To check a solution:

• Substitute the solution back into the original equation.
• Simplify both sides to see if they are equal.
• If both sides match, the solution is correct. If not, re-evaluate your steps.

The beauty of checking solutions is that it removes any guesswork from solving an equation.
It gives you peace of mind knowing that your answer is accurate.

Isolation of Variables

Isolating the variable is a key step in solving linear equations. It means getting the unknown variable, often represented as 'x', 'y', or another letter, by itself on one side of the equation.
To isolate a variable effectively:

• Perform operations to both sides of the equation that help you clean up terms not attached to the variable.
• Use addition, subtraction, multiplication, or division strategically to move terms across the equals sign.

In the given step-by-step solution, adding 6 to both sides was the first move to isolate the variable term.
Tackling each term one at a time makes the process manageable and less prone to mistakes.

Equation Simplification

Equation simplification is about making equations easier to solve by taking them down to their simplest form. This involves combining like terms and reducing equations so that solving them becomes straightforward.
In the given exercise:

• After adding 6 to both sides, the equation \(-3a = 15\) is much simpler to work with than the original.

Remember:

• Always perform the same operation on both sides to maintain the equation's balance.
• Simplification may involve several steps, so proceed methodically.

Simplifying can often be the difference between a confusing equation and an easily solvable one.

Division in Algebra

Division in algebra comes in handy when you’re down to the final step of isolating a variable. When a variable is multiplied by a coefficient, division allows you to solve that multiplication by finding the reciprocal operation.
For instance, in the equation \(-3a = 15\), dividing both sides by \(-3\) gives you the solution for \('a'\).
Here's how:

• Divide each term by the coefficient of the variable (in this case, -3).
• Simplify the fraction if possible to get your final answer.

This division reveals that \('a' = -5\) once simplified.
Practicing these steps regularly will improve your proficiency in handling and solving linear equations with skill.