Problem 61 - Algebra A Combined Function

Question

\(-\frac{3}{7} p=-2\)

Step-by-Step Solution

Answer

\(p = \frac{14}{3}\)

Step 1: Understand the Equation

We have the equation \(-\frac{3}{7} p = -2\). Our goal is to solve for the variable \(p\). This means we want to find the value of \(p\) that makes this equation true.

Step 2: Isolate the Variable \(p\)

To solve for \(p\), we need to get \(p\) by itself on one side of the equation. Currently, \(p\) is being multiplied by \(-\frac{3}{7}\). To isolate \(p\), we need to divide both sides of the equation by \(-\frac{3}{7}\). This gives us: \[ p = \frac{-2}{-\frac{3}{7}} \]

Step 3: Simplify the Expression

Now let's simplify \(\frac{-2}{-\frac{3}{7}}\). Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we have: \[ p = -2 \times \left(-\frac{7}{3}\right) \]

Step 4: Multiply the Numbers

Now we perform the multiplication: \[ p = -2 \times \left(-\frac{7}{3}\right) = \frac{14}{3} \] The negatives cancel each other out, so the result is positive.

Key Concepts

Solving Equations

Solving equations in algebra is about finding the value of a variable that makes the equation true. Equations are like a balance scale. To keep the balance, whatever operation we perform on one side of the equation, we must also perform on the other side. For instance, if we add a number to one side, we must add the same number to the other. This keeps both sides equal.

When approaching the solution of \(-\frac{3}{7} p = -2\),notice that our unknown variable is \(p\). Our aim is to find what value of \(p\) satisfies this equation. Solving equations often involves reversing operations around the variable to find its value. Operations like addition are reversed by subtraction, and multiplication by division. This understanding will guide us through to the solution of the equation.

Isolation of Variables

Isolating the variable means getting the variable you are solving for alone on one side of the equation. Essentially, we're trying to "clear away" everything else so that the variable is all by itself. It’s like peeling away layers to reveal the variable.

• Identify what is attached to the variable: In the equation \(-\frac{3}{7} p = -2\), \(p\) is multiplied by \(-\frac{3}{7}\).
• To isolate \( p \), perform the opposite operation: Since \( p \) is multiplied, we divide by \(-\frac{3}{7}\).
• The operation: Dividing both sides by \(-\frac{3}{7}\) results in \[ p = \frac{-2}{-\frac{3}{7}} \].

This process ensures that \(p\) is isolated, making it easier to determine the value of \(p\). By isolating the variable, the problem becomes less complex and prepares us for the final steps of solving.

Multiplying Fractions

Multiplying fractions may seem intimidating, but it's a straightforward process once you understand the steps. Remember, multiplying fractions doesn't require common denominators, making it simpler in some aspects than adding or subtracting them.

• When dividing by a fraction, instead multiply by its reciprocal: In our equation \(\frac{-2}{-\frac{3}{7}}\), we change \(-\frac{3}{7}\) to its reciprocal, \(-\frac{7}{3}\).
• Next, multiply: We have \[-2 \times \left(-\frac{7}{3}\right)\].
• Calculate: Multiply the numerators together and the denominators together: \(2 \times 7 = 14\) and \(1 \times 3 = 3\), resulting in \(\frac{14}{3}\).
• Notice: Two negatives make a positive. In mathematical terms, multiplying two negative numbers gives a positive product.

By understanding how to handle fractions through reciprocal and multiplication, we can successfully solve equations involving fractions, just like we did here.