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]
