Simplify the following expression:
$\left(\frac{1}{7} x+\frac{3}{8}\right)+\left(\frac{2}{9} x-\frac{1}{8}\right)$
A. $-\frac{5}{63} x+\frac{1}{2}$
B. $\frac{23}{63} x+\frac{1}{2}$
C. $\frac{3}{16} x+\frac{1}{4}$
D. $\frac{23}{63} x+\frac{1}{4}$
Answer (2)
Combine the x terms: 7 1 x + 9 2 x = 63 23 x .
Combine the constant terms: 8 3 − 8 1 = 4 1 .
Combine the simplified terms: 63 23 x + 4 1 .
The simplified expression is 63 23 x + 4 1 .
Explanation
Understanding the Expression We are given the expression ( 7 1 x + 8 3 ) + ( 9 2 x − 8 1 ) and we want to simplify it by combining like terms. This means we need to add the coefficients of the x terms and add the constant terms separately.
Combining x Terms First, let's combine the x terms. We have 7 1 x + 9 2 x . To add these, we need a common denominator, which is 7 × 9 = 63 . So we rewrite the fractions as 7 1 = 63 9 and 9 2 = 63 14 . Therefore, 7 1 x + 9 2 x = 63 9 x + 63 14 x = 63 9 + 14 x = 63 23 x .
Combining Constant Terms Next, let's combine the constant terms. We have 8 3 − 8 1 = 8 3 − 1 = 8 2 = 4 1 .
Combining Simplified Terms Now, we combine the simplified x term and the simplified constant term: 63 23 x + 4 1 .
Selecting the Correct Answer Finally, we compare our simplified expression 63 23 x + 4 1 with the given options. Option D is 63 23 x + 4 1 , which matches our simplified expression. Therefore, the correct answer is D.
Examples
Imagine you're baking a cake and need to combine different amounts of flour and sugar. If you have 7 1 cup of flour and add another 9 2 cup, you need to find a common denominator to determine the total amount of flour. Similarly, if you have 8 3 cup of sugar and use 8 1 cup for frosting, you subtract to find the remaining sugar. This problem demonstrates how combining like terms helps in everyday situations involving measurements and quantities.
The expression ( 7 1 x + 8 3 ) + ( 9 2 x − 8 1 ) simplifies to 63 23 x + 4 1 . The correct answer option is D.
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