Problem 96 - Elementary Algebra

Question

Simplify each algebraic expression by combinang similar terms. $$-\frac{9}{10} x-\frac{3}{14} y+\frac{2}{25} x+\frac{5}{21} y$$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{41}{50}x + \frac{1}{42}y\).

Step 1: Identify Like Terms

In the given expression, identify the terms containing the same variable. Like terms are terms that have the same variable raised to the same power. The terms \(-\frac{9}{10}x\) and \(\frac{2}{25}x\) are like terms because they both involve the variable \(x\). Similarly, the terms \(-\frac{3}{14}y\) and \(\frac{5}{21}y\) are like terms because they both involve the variable \(y\).

Step 2: Combine Like Terms for x

Add the coefficients of the like terms involving \(x\). The coefficients are \(-\frac{9}{10}\) and \(\frac{2}{25}\). To combine them, find a common denominator. The least common denominator (LCD) of 10 and 25 is 50. Rewrite each fraction:\[-\frac{9}{10} = -\frac{45}{50},\]\[\frac{2}{25} = \frac{4}{50}.\]Then, combine:\[-\frac{45}{50}x + \frac{4}{50}x = -\frac{41}{50}x.\]

Step 3: Combine Like Terms for y

Add the coefficients of the like terms involving \(y\). The coefficients are \(-\frac{3}{14}\) and \(\frac{5}{21}\). To combine them, find a common denominator. The least common denominator (LCD) of 14 and 21 is 42. Rewrite each fraction:\[-\frac{3}{14} = -\frac{9}{42},\]\[\frac{5}{21} = \frac{10}{42}.\]Then, combine:\[-\frac{9}{42}y + \frac{10}{42}y = \frac{1}{42}y.\]

Step 4: Write Simplified Expression

Combine the simplified terms from Step 2 and Step 3. The simplified expression is:\[-\frac{41}{50}x + \frac{1}{42}y.\]This is the final simplified form of the given algebraic expression.

Key Concepts

Combining Like Terms

When you're simplifying algebraic expressions, combining like terms is a critical step. It's important to understand that like terms are terms in an expression that have the same variable and the same exponent. For instance, in the expression \[-\frac{9}{10}x - \frac{3}{14}y + \frac{2}{25}x + \frac{5}{21}y,\]\(-\frac{9}{10}x\) and \(\frac{2}{25}x\) are like terms because they both contain the variable \(x\) to the first power. Similarly, \(-\frac{3}{14}y\) and \(\frac{5}{21}y\) are like terms because they contain the variable \(y\). By combining these like terms, you simplify the expression and make it much easier to work with. Here's how you can do that:

• Identify like terms by finding terms with the same variable and exponent.
• Add or subtract the coefficients of the like terms.

Combining like terms reduces the complexity of equations and expressions, making further operations and calculations more straightforward.

Least Common Denominator

The least common denominator (LCD) plays a crucial role when you're dealing with fractions, especially when combining like terms. To combine fractions smoothly, both fractions need to have the same denominator. This is where the LCD comes in. It is the smallest number that each of the denominators can divide into without leaving a remainder.In the problem:

• The terms with variable \(x\) have coefficients \(-\frac{9}{10}\) and \(\frac{2}{25}\). Here, the denominators are 10 and 25, and the LCD is 50.
• For the terms with variable \(y\), the coefficients \(-\frac{3}{14}\) and \(\frac{5}{21}\) have denominators 14 and 21. The LCD for these is 42.

Once you find the LCD, you rewrite each fraction with this common denominator, allowing you to add or subtract them. This step is essential because without the same denominator, the fractions can't be directly combined.

Fraction Addition

Adding fractions can often seem tricky, especially when they have different denominators. But once you've identified the least common denominator, the process becomes much simpler. In this exercise, you first need to rewrite each fraction so that they share a common denominator.Here's the step-by-step process:

• Convert each fraction to an equivalent fraction with the LCD as the new denominator.
• For example, convert \(-\frac{9}{10}\) to \(-\frac{45}{50}\) and \(\frac{2}{25}\) to \(\frac{4}{50}\) when dealing with the \(x\) terms.
• Add or subtract the numerators while keeping the common denominator.

For instance, adding the like terms of \(x\):\[-\frac{45}{50} + \frac{4}{50} = -\frac{41}{50}.\]This process also applies to terms involving \(y\), where \(-\frac{9}{42} + \frac{10}{42} = \frac{1}{42}\). This method ensures that each like term is reduced to its simplest form, ready for further algebraic manipulation.