Add the expressions [tex]$4-\frac{2}{3}b+\frac{1}{4}a$[/tex] and [tex][tex]$\frac{1}{2}a+\frac{1}{6}b-7$[/tex][/tex]. What is the simplified sum?

A. [tex]-\frac{1}{6}a+\frac{5}{12}b-3[/tex]
B. [tex]\frac{3}{4}a-\frac{1}{2}b-3[/tex]
C. [tex]\frac{3}{4}a+\frac{5}{6}b-3[/tex]
D. [tex]\frac{5}{12}a-\frac{1}{6}b-3[/tex]



Answer :

To add the expressions [tex]\(4 - \frac{2}{3}b + \frac{1}{4}a\)[/tex] and [tex]\(\frac{1}{2}a + \frac{1}{6}b - 7\)[/tex], we need to combine the like terms of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and the constants separately.

Starting with the like terms for [tex]\(a\)[/tex]:
- First expression: [tex]\(\frac{1}{4}a\)[/tex]
- Second expression: [tex]\(\frac{1}{2}a\)[/tex]

Next, we add them together:
[tex]\[ \frac{1}{4}a + \frac{1}{2}a = \frac{1}{4}a + \frac{2}{4}a = \frac{3}{4}a \][/tex]

Now, consider the like terms for [tex]\(b\)[/tex]:
- First expression: [tex]\(-\frac{2}{3}b\)[/tex]
- Second expression: [tex]\(\frac{1}{6}b\)[/tex]

Next, we add them together:
[tex]\[ -\frac{2}{3}b + \frac{1}{6}b = -\frac{4}{6}b + \frac{1}{6}b = -\frac{3}{6}b = -\frac{1}{2}b \][/tex]

Finally, consider the constant terms:
- First expression: [tex]\(4\)[/tex]
- Second expression: [tex]\(-7\)[/tex]

Next, we add them together:
[tex]\[ 4 + (-7) = 4 - 7 = -3 \][/tex]

Combining all the results, we get:
[tex]\[ \frac{3}{4}a - \frac{1}{2}b - 3 \][/tex]

Therefore, the simplified sum of the given expressions is:
[tex]\[ \boxed{\frac{3}{4}a - \frac{1}{2}b - 3} \][/tex]