Problem 12 - Prealgebra

Question

Solve each equation. Check your solution. $$\frac{2}{3} p=-22$$

Step-by-Step Solution

Answer

The solution is \( p = -33 \).

Step 1: Understand the Equation

We start with the equation \( \frac{2}{3} p = -22 \). This means \( p \) has been multiplied by \( \frac{2}{3} \), and the product is \( -22 \). Our goal is to isolate \( p \) to solve for its value.

Step 2: Isolate \( p \)

To solve for \( p \), we need to eliminate the fraction \( \frac{2}{3} \). We do this by multiplying both sides of the equation by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). Thus, we multiply both sides:\[\frac{3}{2} \times \frac{2}{3} p = \frac{3}{2} \times (-22)\]This simplifies to:\[p = \frac{3}{2} \times (-22)\]

Step 3: Simplify the Calculation

Now, perform the multiplication on the right side of the equation:\[p = \frac{3}{2} \times (-22) = -33\]Hence, we find that \( p = -33 \).

Step 4: Verify the Solution

To verify this solution, substitute \( p = -33 \) back into the original equation:\[\frac{2}{3} \times (-33) = -22\]Simplifying the left-hand side:\[\frac{2}{3} imes (-33) = -\frac{66}{3} = -22\]Since both sides of the equation are equal, \( p = -33 \) is indeed the correct solution to the equation.

Key Concepts

Prealgebra

Prealgebra lays the foundation for understanding algebra by dealing with basic mathematical concepts such as arithmetic operations and understanding variables. It is the gateway to more complex math topics and begins with simple equations that contain an unknown value referred to as a variable.

In the context of this problem, we are given the equation \( \frac{2}{3} p = -22 \). This equation tells us that "two-thirds of \( p \) equals -22." Our task in prealgebra is to solve such equations to find the value of this unknown variable, \( p \). Solving an equation means finding the particular value of the variable that makes the equation true.

This involves understanding how to manipulate the equation using various arithmetic operations to isolate the variable on one side, which ultimately tells us its value. This sets the stage for solving more complex equations in higher-level algebra.

Fractions

Fractions are numbers that represent parts of a whole. They consist of a numerator (top number) and a denominator (bottom number). In the equation \( \frac{2}{3} p = -22 \), the fraction \( \frac{2}{3} \) indicates that \( p \) is being multiplied by two-thirds.

Dealing with fractions in equations often requires using their reciprocal. The reciprocal of a fraction is found by swapping its numerator and denominator. So the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).

When solving equations involving fractions, we multiply both sides by the reciprocal. This "undoes" the fraction part of the equation by effectively cancelling it out, allowing the variable to be isolated. For our equation:

• Multiply both sides by \( \frac{3}{2} \): \( \frac{3}{2} \times \frac{2}{3} p = \frac{3}{2} \times (-22) \)

The left side simplifies to \( p \) because \( \frac{3}{2} \times \frac{2}{3} = 1 \). This process effectively removes the fraction, making the equation easier to solve.

Equation Verification

Once you find a solution to an equation, it's crucial to verify it, ensuring it's correct. Verification involves substituting the found value back into the original equation to check if the equation holds true.

Let's verify \( p = -33 \) as the solution to our equation \( \frac{2}{3} p = -22 \):

• Substitute \( -33 \) for \( p \): \( \frac{2}{3} \times (-33) \)
• Calculate: \( \frac{2}{3} \times (-33) = -\frac{66}{3} = -22 \)

If both sides of the equation equal \(-22\), the solution is verified. Verification provides a critical check on our process. Ensuring the solution is correct builds confidence for more complex equation-solving challenges later.