Simplify these expressions.
To simplify the given algebraic expressions, we identify and combine like terms. Like terms are terms that have the exact same variables raised to the same exponents.
For expression (a), \(-8p + (-9p) + (-2p)\), the variable is \(p\). Since all terms share the same variable, they are like terms and can be combined by adding their numerical coefficients.
For expression (b), \(-3jk + 5jk + (-6jk)\), the variables are \(jk\). Since all terms share the same variables, they are like terms and can also be combined by adding their numerical coefficients.
To simplify \(-8p + (-9p) + (-2p)\), we use the distributive property to factor out the common variable \(p\):
\[-8p + (-9p) + (-2p) = (-8 - 9 - 2)p\]
Now, we perform the addition of the numerical coefficients:
\[-8 - 9 - 2 = -17 - 2 = -19\]
Thus, the simplified expression for (a) is \(-19p\).
To simplify \(-3jk + 5jk + (-6jk)\), we use the distributive property to factor out the common variables \(jk\):
\[-3jk + 5jk + (-6jk) = (-3 + 5 - 6)jk\]
Now, we perform the addition of the numerical coefficients:
\[-3 + 5 - 6 = 2 - 6 = -4\]
Thus, the simplified expression for (b) is \(-4jk\).
In conclusion, after combining the coefficients for each set of like terms, we have found that:
The final simplified expressions are \(-19p\) and \(-4jk\).
Combining like terms is essential in algebra for simplifying complex equations. For example, if you are calculating the total cost of items where \(p\) represents the price of a specific type of fruit, and you bought several sets of that fruit, you combine the terms to get the total cost quickly. Similarly, if \(j\) and \(k\) represent dimensions of a rectangular area, simplifying terms involving \(jk\) helps in calculating the total area of multiple sections with the same dimensional variables.