Problem 95 - Elementary Algebra

Question

Simplify each algebraic expression by combinang similar terms. $$\frac{2}{9} x+\frac{5}{12} y-\frac{7}{15} x-\frac{13}{15} y$$

Step-by-Step Solution

Answer

\(-\frac{11}{45}x - \frac{9}{20}y\)

Step 1: Identify Like Terms

In the expression \( \frac{2}{9} x+\frac{5}{12} y-\frac{7}{15} x-\frac{13}{15} y \), identify terms with the same variable. Terms with \( x \) are \( \frac{2}{9}x \) and \(-\frac{7}{15}x \). Terms with \( y \) are \( \frac{5}{12}y \) and \(-\frac{13}{15}y \).

Step 2: Combine Like Terms with 'x'

To combine \( \frac{2}{9}x \) and \(-\frac{7}{15}x \), find a common denominator. The least common multiple of 9 and 15 is 45. Convert both fractions to have this denominator: \( \frac{2}{9}x = \frac{10}{45}x \) and \( -\frac{7}{15}x = -\frac{21}{45}x \). Combine the fractions: \( \frac{10}{45}x - \frac{21}{45}x = -\frac{11}{45}x \).

Step 3: Combine Like Terms with 'y'

To combine \( \frac{5}{12}y \) and \(-\frac{13}{15}y \), again find a common denominator. The least common multiple for 12 and 15 is 60. Convert both fractions: \( \frac{5}{12}y = \frac{25}{60}y \) and \( -\frac{13}{15}y = -\frac{52}{60}y \). Combine the fractions: \( \frac{25}{60}y - \frac{52}{60}y = -\frac{27}{60}y \). Simplify \(-\frac{27}{60}y\) to \(-\frac{9}{20}y\).

Step 4: Combine Results

The expression after combining similar terms is \(-\frac{11}{45}x - \frac{9}{20}y\).

Key Concepts

Combining Like Terms

When dealing with algebraic expressions, the first step is often to combine like terms. Like terms are terms that have the same variable raised to the same power. This means you can add or subtract their coefficients while keeping the variable part unchanged. For example, in the expression \( \frac{2}{9}x + \frac{5}{12}y - \frac{7}{15}x - \frac{13}{15}y \), identify terms with the same variable.

• Terms with \( x \): \( \frac{2}{9}x \) and \( -\frac{7}{15}x \)
• Terms with \( y \): \( \frac{5}{12}y \) and \( -\frac{13}{15}y \)

By combining terms with the same variables, you streamline the expression, significantly simplifying your calculations. It’s important to handle signs correctly - if the term is negative, include the sign when summing up the coefficients.

Least Common Multiple

The least common multiple (LCM) plays a crucial role in combining terms with fractions quickly and accurately. When different fractions have unlike denominators, you cannot directly add or subtract them. Instead, you must find a common denominator, and that’s where the least common multiple comes into play.To combine terms like \( \frac{2}{9}x \) and \( -\frac{7}{15}x \), you first find the least common multiple of 9 and 15. The LCM here is 45. By converting each fraction to have 45 as the denominator:

• \( \frac{2}{9}x \) becomes \( \frac{10}{45}x \)
• \( -\frac{7}{15}x \) becomes \( -\frac{21}{45}x \)

This similar process applies to terms with \( y \). With 12 and 15 having an LCM of 60, convert the fractions accordingly so that they share a common base, allowing straightforward addition or subtraction.

Simplifying Expressions

Once you have combined like terms using the least common multiple, the next step is simplification. Simplifying an expression means to make it as neat and compact as possible. After aligning the denominators and combining like terms, check if the result can be further simplified.For instance, combining the \( x \) terms, the expression becomes \( -\frac{11}{45}x \). Look at the \( y \) terms: initially combined to \( \frac{25}{60}y - \frac{52}{60}y = -\frac{27}{60}y \). Simplify it further to \( -\frac{9}{20}y \).To simplify, ensure that all fractions are in their lowest terms:

• Check the greatest common divisor (GCD) of the numerator and denominator.
• Divide both numerator and denominator by the GCD.

This results in the final simplified expression \( -\frac{11}{45}x - \frac{9}{20}y \), concise and easy to interpret or use in additional calculations.