Problem 68 - Intermediate Algebra

Question

Solve each proportion. $$ \frac{-6}{m}=\frac{m}{-6} $$

Step-by-Step Solution

Answer

The solutions are \( m = 6 \) and \( m = -6 \).

Step 1: Cross Multiply to Clear Fractions

Start by using the cross-multiplication method to eliminate the fractions in the proportion. This means multiplying the numerator of each fraction by the denominator of the other:\[ -6 imes (-6) = m imes m \] which simplifies to \[ 36 = m^2 \]

Step 2: Take the Square Root

Next, solve the equation for \( m \) by taking the square root of both sides. This will give you two possible solutions since both a positive and negative number squared will give the same result:\[ m = \sqrt{36} \quad \text{or} \quad m = -\sqrt{36} \] Thus, \( m = 6 \) or \( m = -6 \).

Step 3: Check Your Solutions

Verify that both solutions satisfy the original proportion by substituting them back into the equation:For \( m = 6 \): \[ \frac{-6}{6} = \frac{6}{-6} \] which simplifies to \(-1 = -1\), so it holds true.For \( m = -6 \): \[ \frac{-6}{-6} = \frac{-6}{-6} \] which simplifies to \(1 = 1\), so it also holds true.

Key Concepts

Cross Multiplication

In solving proportions, cross multiplication is a useful technique. It helps to eliminate fractions by multiplying across the equals sign. Picture each side of the proportion as a fraction. Cross multiplication means taking the numerator of one fraction and multiplying it by the denominator of the other. For instance, consider the proportion \( \frac{-6}{m} = \frac{m}{-6} \). To cross multiply, you will perform the following operation: multiply \(-6\) (numerator of the first fraction) with \(-6\) (denominator of the second fraction). Simultaneously, multiply \(m\) (numerator of the second fraction) with \(m\) (denominator of the first fraction). This results in the equation \(-6 \times (-6) = m \times m\), which simplifies to \(36 = m^2\).

• Cross multiplication removes fractions.
• Makes solving proportions straightforward.

Square Roots

After cross multiplying, often you'll end up with a quadratic equation like \(36 = m^2\). To solve for \(m\), the next step is to take the square root of both sides of the equation. The square root operation helps us find the original number which, when squared, gives the result. It is important to remember that squaring both positive and negative numbers results in the same value. Therefore, when solving \(m^2 = 36\), the potential solutions are both \(m = \sqrt{36}\) and \(m = -\sqrt{36}\). Thus, you get two possible values: \(m = 6\) or \(m = -6\).

• Always consider both positive and negative results of a square root.
• Square roots can lead to two solutions in quadratic equations.

Checking Solutions

Checking solutions is an essential step in verifying correctness in calculations. After deriving potential values for \(m\), substitute each one back into the original proportion equation to verify if they satisfy it. Consider \(m = 6\): substitute it back into \( \frac{-6}{m} = \frac{m}{-6} \). The equation becomes \( \frac{-6}{6} = \frac{6}{-6} \), which simplifies to \(-1 = -1\), confirming it as a valid solution. Now test \(m = -6\): the equation becomes \( \frac{-6}{-6} = \frac{-6}{-6} \), or \(1 = 1\), which is also valid. Therefore, both solutions check out as true.

• Substituting back into the original equation is crucial.
• Helps affirm if the solutions are indeed correct.