Problem 54 - Prealgebra

Question

Solve each equation. Check your solution. $$\frac{3}{7} r=-9$$

Step-by-Step Solution

Answer

The solution is \( r = -21 \).

Step 1: Isolate the Variable

To solve the equation \( \frac{3}{7} r = -9 \), we want to isolate the variable \( r \). To do this, multiply both sides of the equation by the reciprocal of \( \frac{3}{7} \), which is \( \frac{7}{3} \). This yields:\[ r = -9 \times \frac{7}{3} \]

Step 2: Simplify the Expression

Now, calculate the right-hand side:\( -9 \times \frac{7}{3} = -9 \times \frac{7}{3} \) can be simplified to:\[ r = -63 / 3 \]Divide -63 by 3:\[ r = -21 \]

Step 3: Check Your Solution

Substitute \( r = -21 \) back into the original equation to verify:\[ \frac{3}{7}(-21) = -9 \]First, calculate:\( \frac{3}{7} \times -21 = -3 \times 3 = -9 \)Since both sides of the equation are equal, the solution is correct.

Key Concepts

Isolate the Variable

When solving equations, one important step is to isolate the variable you are trying to find. This means getting the variable by itself on one side of the equation. In our given equation \( \frac{3}{7} r = -9 \), the variable is \( r \). To isolate \( r \), we need to remove the fraction \( \frac{3}{7} \) that is multiplied by it.

The trick to isolate a variable in this kind of situation is to use the reciprocal of the fraction. The reciprocal of \( \frac{3}{7} \) is \( \frac{7}{3} \), which you get by flipping the numerator and the denominator. By multiplying both sides of the equation by this reciprocal, you effectively cancel out the fraction, leaving \( r \) alone on one side of the equation:

\[ r = -9 \times \frac{7}{3} \]

• The multiplication of \(-9\) and \(\frac{7}{3}\) is straightforward. Multiply \(-9\) by \(7\) and then divide the product by \(3\).

By following these steps, you successfully isolate the variable on one side of the equation.

Checking Solutions

After you have found a solution to an equation, it's vital to check your work. This involves substituting the solution back into the original equation to see if it holds true. In our example, we found \( r = -21 \). To ensure this is the correct solution, we substitute \( -21 \) back into the original equation:

\[ \frac{3}{7}(-21) = -9 \]

Calculate the left side of the equation:

• First, multiply \(3\) by \(-21\), which gives \(-63\).
• Then, divide \(-63\) by \(7\), resulting in \(-9\).

Since the left side of the equation is equal to \(-9\), which is the same as the right side of the equation, your solution \( r = -21 \) is correct. Checking your solutions helps catch any mistakes and confirms your answer.

Fractions in Equations

Dealing with fractions in equations can seem tricky, but it becomes easier with practice. Fractions often appear in equations and solving them requires understanding how fractions work. In the equation \( \frac{3}{7} r = -9 \), \( \frac{3}{7} \) is a fraction that needs to be managed to solve for \( r \).

When a variable is part of a fraction, the key is to eliminate the fraction. This can be done by multiplying by its reciprocal. For \( \frac{3}{7} \), the reciprocal is \( \frac{7}{3} \), and multiplying it by both sides of the equation clears the fraction from one side:

\[ r = -9 \times \frac{7}{3} \]

Handling fractions correctly means:

• Knowing how to find reciprocals.
• Understanding multiplication and division of fractions.
• Simplifying fractions when possible.

Mastering these skills will enable you to efficiently solve equations that contain fractions.