Problem 12 - Introductory Algebra for College Students

Question

Solve each equation in using the multiplication property of equality. Be sure to check your proposed $$-8 x=6$$

Step-by-Step Solution

Answer

The solution to the equation \(-8x = 6\) is \(x = -\frac{3}{4}\).

Step 1: Understand the Problem

We are given the equation \(-8x = 6\). The task is to find the value of \(x\) by isolating it on one side, using the multiplication property of equality.

Step 2: Divide by -8 on both sides

We can divide by -8 on both sides in order to isolate \(x\). The multiplication property of equality allows us to do this. Thus, the equation becomes \(x = -\frac{6}{8}\).

Step 3: Simplify

We simplify the right side of the equation by reducing the fraction to the lowest term. This gives us \(x = -\frac{3}{4}\).

Step 4: Check the solution

We check the solution by inserting it back into the original equation. Substituting \(x = -\frac{3}{4}\) into \(-8x = 6\), we have \(-8(-\frac{3}{4}) = 6\), and by simplification, we confirm this statement as true, implying our solution is correct.

Key Concepts

Multiplication Property of Equality

The multiplication property of equality is a fundamental concept in algebra. It states that when you multiply or divide both sides of an equation by the same nonzero number, the equality holds true. This property helps us solve equations by allowing us to keep the equation balanced while isolating the variable on one side.

• In the problem \(-8x = 6\), our goal is to find the value of \(x\).
• We need to get \(x\) alone, so we divide both sides by -8, the coefficient of \(x\).

This gives us the equation \(-\frac{6}{8} = x\). By using this property, we can solve for variables efficiently while maintaining equality.

Checking Solutions

Checking solutions ensures that your derived answer is accurate. After solving an equation, substituting your solution back into the original equation verifies its correctness. This step is crucial because it confirms that no mistakes were made during calculations.

• For our equation, we found \(x = -\frac{3}{4}\).
• By substituting back, \(-8( -\frac{3}{4})\) should equal 6.

When calculated, \(-8 \times -\frac{3}{4} \) indeed simplifies to 6. Therefore, our solution is verified as correct.

Reducing Fractions

Reducing fractions is about simplifying them to their lowest terms. This process involves dividing the numerator and the denominator by their greatest common factor (GCF).

• From our solution \(x = -\frac{6}{8}\), recognizing that 6 and 8 have a common factor is crucial.
• The GCF of 6 and 8 is 2.

Dividing both by their GCF, \(\frac{6}{8}\) becomes \(\frac{3}{4}\). Thus, our simplified answer for the equation is \(x = -\frac{3}{4}\). Simplification not only gives us a clearer answer but also makes further calculations more manageable.