Problem 36 - Prealgebra

Question

Solve each equation. Check your solution. $$-\frac{7}{9} t=-\frac{28}{36}$$

Step-by-Step Solution

Answer

The solution is \(t = 1\).

Step 1: Simplify the Equation

First, simplify the fraction on the right side of the equation. The fraction \(-\frac{28}{36}\) can be simplified by finding the greatest common divisor (GCD) of 28 and 36, which is 4. Divide both the numerator and the denominator by 4, resulting in \(-\frac{28}{36} = -\frac{7}{9}\). The equation now becomes: \[-\frac{7}{9} t = -\frac{7}{9}\].

Step 2: Solve for t

Since both sides of the equation are equal, \(-\frac{7}{9}\) is multiplied by the same value of \(t\) and equals the same fraction, we can conclude that \(t\) must be 1. Therefore, divide both sides by \(-\frac{7}{9}\) to isolate \(t\):\[ t = \frac{-7/9}{-7/9} = 1 \].

Step 3: Check the Solution

To verify the solution, substitute \(t = 1\) back into the original equation. Calculate \(-\frac{7}{9} \times 1\) which gives \(-\frac{7}{9}\). Check if this equals \(-\frac{28}{36}\), which we simplified to \(-\frac{7}{9}\) in Step 1. Since both sides are equal, the solution is confirmed.

Key Concepts

Understanding Greatest Common Divisor

The greatest common divisor (GCD) is a very useful concept in mathematics, especially when working with fractions. The GCD of two numbers is the largest positive integer that divides both numbers without leaving a remainder. It helps us simplify fractions by finding the largest number that can divide both the numerator and the denominator evenly.

For example, in the problem where we simplify the fraction \(-\frac{28}{36}\), finding the GCD of 28 and 36 is crucial. To find the GCD:

• List the factors of 28 (1, 2, 4, 7, 14, 28).
• List the factors of 36 (1, 2, 3, 4, 6, 9, 12, 18, 36).
• Identify the common factors (1, 2, 4).
• The greatest common factor is 4.

So, by dividing both 28 and 36 by 4, we simplify the fraction to \(-\frac{7}{9}\). Understanding how to find the GCD helps you quickly simplify fractions and can make solving equations much easier.

Exploring Fractions

Fractions represent a part of a whole and consist of a numerator and a denominator. The numerator is the top number, showing how many parts we have. The denominator is the bottom number, showing how many equal parts make up a whole.

In our problem, we have fractions like \(-\frac{7}{9}\) and \(-\frac{28}{36}\). It’s important to recognize that fractions can often be simplified if there is a common number that divides both the numerator and the denominator. This simplification often makes them easier to use.

Fractions can be positive or negative, and they follow the same rules as whole numbers. When working with equations, it’s essential to maintain the balance by performing the same operation on both sides, whether multiplying, dividing, or anything else. This ensures the equation remains true, which is a key aspect of solving linear equations.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. This means rewriting the fraction so that the numerator and denominator are as small as possible and have no common factors apart from 1.

Consider the fraction \(-\frac{28}{36}\) from our problem. We simplified it by using the greatest common divisor, which was 4, to reduce it to \(-\frac{7}{9}\):

• Divide 28 by 4 to get 7 (the new numerator).
• Divide 36 by 4 to get 9 (the new denominator).

The fraction \(-\frac{7}{9}\) is its simplest form because there are no common divisors of 7 and 9 other than 1. Simplifying fractions not only makes them easier to work with, but it also gives a clearer understanding of their value, which is especially useful when solving equations or comparing different fractions.