To combine like terms and create an equivalent expression for the given mathematical expression:
[tex]\[
-\frac{2}{3} p + \frac{1}{5} - 1 + \frac{5}{6} p
\][/tex]
we follow these steps:
1. Identify and combine the terms involving [tex]\( p \)[/tex]:
- The terms involving [tex]\( p \)[/tex] are [tex]\( -\frac{2}{3} p \)[/tex] and [tex]\( \frac{5}{6} p \)[/tex].
2. Combine the coefficients of [tex]\( p \)[/tex]:
- Combine the fractions [tex]\( -\frac{2}{3} \)[/tex] and [tex]\( \frac{5}{6} \)[/tex].
3. Finding a common denominator:
- The common denominator between [tex]\( -\frac{2}{3} \)[/tex] and [tex]\( \frac{5}{6} \)[/tex] is 6.
- Rewrite [tex]\( -\frac{2}{3} \)[/tex] as [tex]\( -\frac{4}{6} \)[/tex].
4. Add the fractions:
- Adding [tex]\( -\frac{4}{6} \)[/tex] and [tex]\( \frac{5}{6} \)[/tex]:
[tex]\[
-\frac{4}{6} + \frac{5}{6} = \frac{1}{6}
\][/tex]
- Therefore, the combined [tex]\( p \)[/tex] term is [tex]\( \frac{1}{6} p \)[/tex].
5. Identify and combine the constant terms:
- The constant terms in the expression are [tex]\( \frac{1}{5} \)[/tex] and [tex]\( -1 \)[/tex].
6. Convert -1 to a fraction with the same denominator as [tex]\( \frac{1}{5} \)[/tex]:
- Rewrite [tex]\( -1 \)[/tex] as [tex]\( -\frac{5}{5} \)[/tex].
7. Combine the fractions:
- Adding [tex]\( \frac{1}{5} \)[/tex] and [tex]\( -\frac{5}{5} \)[/tex]:
[tex]\[
\frac{1}{5} - \frac{5}{5} = -\frac{4}{5}
\][/tex]
- Therefore, the combined constant term is [tex]\( -\frac{4}{5} \)[/tex].
8. Combine all the simplified terms:
- The final expression after combining like terms is:
[tex]\[
\frac{1}{6} p - \frac{4}{5}
\][/tex]
So the equivalent expression with combined like terms is:
[tex]\[
\frac{1}{6} p - \frac{4}{5}
\][/tex]
