Problem 16 - Prealgebra

Question

Solve each equation. Check your solution. $$-56=-7 p$$

Step-by-Step Solution

Answer

The solution is \(p = 8\). Substituting back verifies it.

Step 1: Understand the Equation

The given equation is \(-56 = -7p\). Our goal is to solve for \(p\). This means we need to isolate \(p\) on one side of the equation.

Step 2: Isolate the Variable

To solve for \(p\), divide both sides of the equation by \(-7\). This operation will isolate \(p\) on the right side:\[\frac{-56}{-7} = \frac{-7p}{-7}\]This simplifies to:\[8 = p\] or \(p = 8\).

Step 3: Check the Solution

Substitute \(p = 8\) back into the original equation to check if it satisfies the equation:\[-56 = -7(8)\]Calculate the right side:\[-7 imes 8 = -56\].Since both sides are equal, the solution is correct.

Key Concepts

Isolating Variables

When solving an equation like \(-56 = -7p\), our primary goal is to determine the value of the unknown variable, which is \(p\) in this instance. To achieve this, we need to isolate the variable. Isolating the variable means placing \(p\) alone on one side of the equation so we can directly see its value on the other side.Here's how we can isolate the variable in this scenario:

• The equation is given as \(-56 = -7p\).
• Since \(-7p\) means \(-7 \times p\), we can reverse the multiplication through division.
• Therefore, we divide both sides of the equation by \(-7\) to simplify it and solve for \(p\): \[\frac{-56}{-7} = \frac{-7p}{-7}\]

The division cancels out the \(-7\) on the right side, leaving \(p = 8\). This gives us a clearer view of the variable's value. Isolating variables is a critical step in solving equations and helps make complex problems easier to understand.

Equation Verification

Once we have a solution for the equation, it’s essential to verify its validity. Verification ensures that the value of the variable truly satisfies the equation. This step is crucial because minor errors in calculation or logic can lead to incorrect results.In our example, with \(p = 8\):

• Plug \(p = 8\) back into the original equation: \[-56 = -7(8)\]
• Calculate the right side: \(-7 \times 8 = -56\)
• Since the left side \(-56\) matches the calculated right side \(-56\), the solution \(p = 8\) is verified successfully.

Equation verification acts as a reliable check to confirm solution accuracy. By substituting the value back into the original equation, we ensure the problem has been solved correctly. This practice helps build confidence in your solutions and strengthens your understanding of algebraic principles.

Mathematical Operations

Understanding and choosing the correct mathematical operation is key when handling equations. In the equation \(-56 = -7p\), our task is to decide which operations will help isolate the variable and solve the problem.Operations in Equations:

• Identify that \(-7p\) means \(-7 \times p\). Here, multiplication is in use.
• To undo this multiplication, we apply the inverse operation, which is division. Therefore, we divide by \(-7\) to eliminate the coefficient of \(p\).
• Executing this division:\[\frac{-56}{-7} = \frac{-7p}{-7} \] streamlines the equation to show \(p = 8\).

Knowing how to correctly apply operations like addition, subtraction, multiplication, and division is foundational in solving equations. Each operation has an "inverse" or opposite operation that can help 'undo' it, making certain that you choose the right path to simplifying or solving equations efficiently.