Problem 62
Question
Simplify each expression. \(-\frac{5}{6}+8 x+\frac{1}{6} x-7-\frac{7}{6}\)
Step-by-Step Solution
Verified Answer
\(\frac{49}{6}x - 9\)
1Step 1 - Identify Like Terms
Separate the like terms in the given expression. The like terms in the expression \(-\frac{5}{6}+8x+\frac{1}{6}x-7-\frac{7}{6}\) are constants and terms with \(x\).
2Step 2 - Combine Constants
Add the constant terms together: \(-\frac{5}{6} - 7 - \frac{7}{6} \). To do this, first convert all constants to have a common denominator:\( -\frac{5}{6} = -\frac{5}{6} \),\( -7 = -\frac{42}{6} \), and \(-\frac{7}{6} = -\frac{7}{6}\).Adding these together: \(-\frac{5}{6} - \frac{42}{6} - \frac{7}{6} = -\frac{54}{6} = -9\).
3Step 3 - Combine \(x\) Terms
Combine the \(x\) terms: \(8x + \frac{1}{6}x\). This gives us: \(\frac{48}{6}x + \frac{1}{6}x = \frac{49}{6}x\).
4Step 4 - Rewrite the Simplified Expression
Combine the simplified constants and \(x\) terms to get the final expression: \(\frac{49}{6}x - 9\).
Key Concepts
Combining Like TermsCommon DenominatorConstants and Variables
Combining Like Terms
In algebra, one of the essential steps in simplifying expressions is combining like terms.
Like terms are terms that have the same variable raised to the same power. This means only the coefficients (the numerical part) can vary.
For instance, in the expression \(8x + \frac{1}{6}x\), both terms are 'like' because they both have the variable \(x\).
To combine them, simply add their coefficients: \(8 + \frac{1}{6} = \frac{48}{6} + \frac{1}{6} = \frac{49}{6}\).
Here are a few tips to help you combine like terms:
Like terms are terms that have the same variable raised to the same power. This means only the coefficients (the numerical part) can vary.
For instance, in the expression \(8x + \frac{1}{6}x\), both terms are 'like' because they both have the variable \(x\).
To combine them, simply add their coefficients: \(8 + \frac{1}{6} = \frac{48}{6} + \frac{1}{6} = \frac{49}{6}\).
Here are a few tips to help you combine like terms:
- Look for terms with identical variables and exponents.
- Add or subtract the coefficients while keeping the variable and exponent the same.
- Always check your result to make sure all like terms are combined.
Common Denominator
When dealing with fractions in algebra, finding a common denominator is crucial.
A common denominator makes it easier to add or subtract fractions because it converts them to have the same bottom part.
For example, in the expression \(-\frac{5}{6} - 7 - \frac{7}{6}\), the terms \(-\frac{5}{6}\) and \(-\frac{7}{6}\) already have a denominator of 6.
To combine these fractions with \(-7\), we convert \(-7\) to have a denominator of 6: \(-7 = -\frac{42}{6}\).
Combining these gives us: \(-\frac{5}{6} - \frac{42}{6} - \frac{7}{6} = -\frac{54}{6} = -9\).
Here are some tips on finding common denominators:
A common denominator makes it easier to add or subtract fractions because it converts them to have the same bottom part.
For example, in the expression \(-\frac{5}{6} - 7 - \frac{7}{6}\), the terms \(-\frac{5}{6}\) and \(-\frac{7}{6}\) already have a denominator of 6.
To combine these fractions with \(-7\), we convert \(-7\) to have a denominator of 6: \(-7 = -\frac{42}{6}\).
Combining these gives us: \(-\frac{5}{6} - \frac{42}{6} - \frac{7}{6} = -\frac{54}{6} = -9\).
Here are some tips on finding common denominators:
- Identify the denominators in the fractions you are working with.
- Find the least common multiple (LCM) of these denominators.
- Convert each fraction to an equivalent fraction with the common denominator.
- Perform the addition or subtraction across the numerators.
Constants and Variables
In algebraic expressions, understanding the distinction between constants and variables is vital.
A constant is a fixed number, like -7 or 3.
In contrast, a variable represents an unknown value and is typically shown with letters like \(x\) or \(y\).
For example, in the expression \(8x + \frac{1}{6}x - \frac{5}{6} - 7 - \frac{7}{6}\), \(x\) and \(\frac{1}{6}x\) are variable terms, while \(-\frac{5}{6}, -7,\) and \(-\frac{7}{6}\) are constants.
To simplify, you handle constants and variables separately.
Combine like terms of variables as one group and constants as another.
For example, variable terms: \(8x + \frac{1}{6}x = \frac{49}{6}x\).
Constant terms: \(-\frac{5}{6} - 7 - \frac{7}{6} = -9\).
Some helpful points to remember:
A constant is a fixed number, like -7 or 3.
In contrast, a variable represents an unknown value and is typically shown with letters like \(x\) or \(y\).
For example, in the expression \(8x + \frac{1}{6}x - \frac{5}{6} - 7 - \frac{7}{6}\), \(x\) and \(\frac{1}{6}x\) are variable terms, while \(-\frac{5}{6}, -7,\) and \(-\frac{7}{6}\) are constants.
To simplify, you handle constants and variables separately.
Combine like terms of variables as one group and constants as another.
For example, variable terms: \(8x + \frac{1}{6}x = \frac{49}{6}x\).
Constant terms: \(-\frac{5}{6} - 7 - \frac{7}{6} = -9\).
Some helpful points to remember:
- Identify and separate constants and variables in your expression.
- Combine constants using common denominators if necessary.
- Combine variable terms by adding or subtracting their coefficients.