Problem 95 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{5 x}{-2}-6=-10 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{8}{5}\).

Step 1: Remove Constants on Both Sides

To start solving the equation, we need to eliminate the constant on the left side of the equation. Add 6 to both sides:\[\frac{5x}{-2} - 6 + 6 = -10 + 6\]This simplifies to:\[\frac{5x}{-2} = -4\]

Step 2: Isolate the Variable

Now that we have \(\frac{5x}{-2} = -4\), multiply both sides by \(-2\) to clear the fraction:\[5x = -4 imes -2\]Which simplifies to:\[5x = 8\]

Step 3: Solve for x

Finally, divide both sides by 5 to solve for \(x\):\[x = \frac{8}{5}\]

Step 4: Final Step: Verify Your Solution

Substitute \(x = \frac{8}{5}\) back into the original equation to verify:\[\frac{5\left(\frac{8}{5}\right)}{-2} - 6 = -10\]Simplify inside the fraction:\[\frac{8}{-2} - 6 = -4 - 6 = -10\]The left side equals the right side, confirming our solution is correct.

Key Concepts

Step-by-step solution

Understanding how to solve a linear equation effectively takes practice and patience. Following a step-by-step approach ensures that no steps are overlooked and the correct solution is reached.
In this problem, we aim to find the value of the variable \(x\) that makes the equation true. Our linear equation to solve is \(\frac{5x}{-2} - 6 = -10\).
Here is a simple breakdown of the steps involved:

• Step 1: Move constants - We start by removing the constant \(-6\) from the left side by adding 6 to both sides. This simplifies to \(\frac{5x}{-2} = -4\).
• Step 2: Clear fractions - Multiply both sides by \(-2\) to eliminate the fraction. This results in \(5x = 8\).
• Step 3: Solve for \(x\) - Finally, divide both sides by 5 to isolate \(x\), which gives us \(x = \frac{8}{5}\).

This clear-cut sequence of actions leads us to the solution in a logical manner, ensuring accuracy at each step.

Equation simplification

Simplifying an equation involves manipulating it to make it more manageable while preserving its equality. This enables us to isolate the variable we are solving for.
In the problem \(\frac{5x}{-2} - 6 = -10\), simplification begins with addressing the constants and fractions.

• Addressing constants: The subtraction by 6 on the left side can be canceled by adding 6 to both sides. This removes the constant and focuses on terms involving the variable.
• Handling fractions: The fraction \(\frac{5x}{-2}\) can be confusing, so we eliminate the fraction by multiplying every term by \(-2\). This simplification step transforms the equation into a simpler linear form: \(5x = 8\).

Simple adjustments to the equation like these allow us to focus directly on solving for the variable \(x\), leading to a more straightforward solution process.

Verifying solutions

Once we find a solution, it is vital to ensure it is correct. Verification provides confidence that the actions taken were appropriate and the solution is valid.
After obtaining \(x = \frac{8}{5}\), we substitute it back into the original equation to verify.

• Substitute \(x\) back into the equation \(\frac{5x}{-2} - 6 = -10\) to check if both sides equal.
• In our case, \(\frac{5(\frac{8}{5})}{-2} - 6\) simplifies to \(\frac{8}{-2} - 6\), which is \(-4 - 6 = -10\).

Both sides equal \(-10\), confirming that \(x = \frac{8}{5}\) is indeed the correct solution. This last step is crucial in problem-solving as it assures us of the solution's integrity.