Problem 31 - Prealgebra

Question

Solve each equation. Check your solution. $$-6=\frac{3}{5} a$$

Step-by-Step Solution

Answer

The solution to the equation is \( a = -10 \).

Step 1: Understand the Equation

The given equation is \[-6 = \frac{3}{5}a\]. This is a simple linear equation where the variable \(a\) is multiplied by a fraction.

Step 2: Isolate the Variable

To isolate \(a\), we need to eliminate the fraction. Multiply both sides of the equation by \( \frac{5}{3} \), which is the reciprocal of \( \frac{3}{5} \).\[-6 \times \frac{5}{3} = a\].

Step 3: Perform the Multiplication

Calculate the value of the expression on the left by multiplying:\[-6 \times \frac{5}{3} = -6 \times \frac{5}{3} = -\frac{30}{3} = -10\]. So, \(a = -10\).

Step 4: Check the Solution

Substitute \( a = -10 \) back into the original equation to ensure it satisfies the condition:\[ \frac{3}{5}(-10) = -6 \].Calculate: \[-6 = -6\]. The left side equals the right side, so \( a = -10 \) is a correct solution.

Key Concepts

Understanding Variable Isolation

In solving linear equations, one of the key techniques used is variable isolation. This means rearranging the equation to make the variable stand alone on one side of the equation, often the right side. To isolate the variable in the equation \[-6 = \frac{3}{5}a\],we need to eliminate the fraction that's multiplying the variable. Isolation is crucial because it allows us to clearly identify the value of the variable. It involves reversing any operations affecting the variable by applying inverse operations.Here's how:

• Identify the operation affecting the variable (here, multiplication by \(\frac{3}{5}\)).
• Apply the inverse operation (in this case, multiplication by its reciprocal \(\frac{5}{3}\)).
• Perform the operation equally on both sides to maintain balance.

With these steps, the variable can be isolated and the equation rephrased into a simpler form, leading to the solution.

Fraction Multiplication Made Simple

When dealing with equations that involve fractions, such as\[-6 = \frac{3}{5}a\],understanding how to handle fractions is essential, especially fraction multiplication. In this context:- The fraction \(\frac{3}{5}\) is multiplying the variable \(a\). - To eliminate the fraction, multiply both sides of the equation by the reciprocal of \(\frac{3}{5}\), which is \(\frac{5}{3}\). Here’s a quick guide on fraction multiplication:

• Multiplying fractions involves multiplying the numerators together and the denominators together.
• If you multiply \(-6\) by \(\frac{5}{3}\), it becomes \[-6 \times \frac{5}{3} = -\frac{30}{3} = -10\].
• Always simplify your result if possible.

Neutralising fractions through such multiplication simplifies equations, making it easier to solve for variables.

Equation Solving Techniques

Solving linear equations primarily involves applying techniques to manipulate and simplify the equation until the solution is apparent. Solving the equation\[-6 = \frac{3}{5}a\]involves specific steps:

• First, identify the operations performed on the variable (here, multiplication by a fraction).
• Second, systematically apply inverse operations to achieve a simpler form, such as isolating \(a\).
• Third, simplify any resulting expressions, like \(-\frac{30}{3}\) which further reduces to \(-10\).

Linear equations like this one are straightforward once these fundamental steps are followed. Practice using these techniques can transform complex problems into simple, resolvable ones.

Solution Verification Process

After finding a solution to an equation, verifying the result ensures its correctness and accuracy. Let’s verify our solution of \(a = -10\)for the equation\[-6 = \frac{3}{5}a\].Verification is simply assessing whether substituting our solution back into the original equation holds true:

• Replace \(a\) with \(-10\) in the original equation: \(\frac{3}{5}(-10)\).
• Perform the arithmetic: \(\frac{3}{5} \times -10 = -6\).
• Check if both sides of the equation are equal: \(-6 = -6\).

If both sides of the equation match, the solution is verified. This step is essential to confirm that no errors were made during calculations or algebraic manipulations.