Problem 56 - Prealgebra

Question

Solve each equation. Check your solution. $$-4.8=6 z$$

Step-by-Step Solution

Answer

The solution to the equation is \(z = -0.8\).

Step 1: Identify the equation

The given equation is \(-4.8 = 6z\). Our goal is to solve for \(z\).

Step 2: Isolate the variable

To isolate \(z\), we need to divide both sides of the equation by 6. This operation will allow us to solve for \(z\) in terms of a single variable. Thus, we have \(z = \frac{-4.8}{6}\).

Step 3: Perform the division

Calculate \(z = \frac{-4.8}{6}\). This simplifies to \(z = -0.8\).

Step 4: Check the solution

Substitute \(z = -0.8\) back into the original equation to verify the solution. Compute \(6 \times (-0.8) = -4.8\). Since the left-hand side equals the right-hand side, our solution \(z = -0.8\) is correct.

Key Concepts

Isolating the Variable

In solving equations, one of the key steps is isolating the variable you are trying to solve for. This is crucial because it transforms more complex expressions into simpler terms, allowing you to easily interpret the answer. When we refer to isolating the variable, we mean employing mathematical operations to make the variable appear by itself on one side of the equation.
This often involves inverse operations, such as undoing addition or subtraction with subtraction or addition, and undoing multiplication with division, as seen in our example equation \[-4.8 = 6z\]Here, the variable is "\(z\)" and it is currently multiplied by 6. To isolate \(z\), you would divide both sides of the equation by 6, as division is the inverse operation of multiplication. Performing this on both sides ensures the equality remains unchanged, resulting in \(z = \frac{-4.8}{6}\). Now, with a clear and simple expression, \(z\) stands alone and ready for further examination.

Checking Solutions

After you've solved an equation by isolating the variable, a critical step is to check your solution. This process involves substituting the obtained value back into the original equation to ensure that both sides remain equal.
This verification step confirms the accuracy of your solution. In our example, once we found \(z = -0.8\), we substituted this back into the original equation to check:\[6 \times (-0.8) = -4.8\]It's helpful to write out each substitution clearly, as it provides visual confirmation that the left-hand side correctly matches the original right-hand side. If the calculated value does not satisfy the equation, it's a sign that there might have been an error in the previous steps, highlighting the importance of this verification.

Simplifying Fractions

Simplifying fractions is a fundamental concept in solving equations, which often become necessary once the variable is isolated. In the isolation process, you might end up with a fraction that represents your variable, as in our example:\[z = \frac{-4.8}{6}\]Simplifying a fraction involves expressing it in fewer terms without changing its value. To simplify, divide both the numerator and the denominator by their greatest common divisor (GCD). In this case, you calculate \[\frac{-4.8}{6} = -0.8\]Since -0.8 is already in its simplest decimal form, no further simplification is needed. However, understanding how to convert between fractions and decimals, as well as simplifying them, enhances your mathematical skills and ensures the results are easier to understand and verify.