Problem 317 - Intermediate Algebra

Question

In the following exercises, simplify. $$ 6.8 p+9.14 q+(-4.37 p)+(-0.88 q) $$

Step-by-Step Solution

Answer

2.43p + 8.26q

Step 1 - Identify Like Terms

Group the terms with the same variables together. In this case, group the terms with 'p' and the terms with 'q'.

Step 2 - Combine the 'p' Terms

Add the coefficients of the 'p' terms: group the coefficients of 'p': [6.8p + (-4.37p)].

Step 3 - Combine the 'q' Terms

Add the coefficients of the 'q' terms: [ 9.14q + (-0.88q)].

Step 4 - Simplify the Grouped Terms

Perform the arithmetic operations for the combined coefficients:6.8 - 4.37 for the 'p' terms and 9.14 - 0.88 for the 'q' terms.

Step 5 - Write the Final Expression

After simplifying, you will get the final expression as: 2.43p + 8.26q.

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, an essential step is combining like terms. Like terms are terms that have the same variable raised to the same power. This means their variable part is identical.
For instance, in the expression 6.8p + 9.14q + (-4.37p) + (-0.88q), we identify two pairs of like terms:

• 6.8p and -4.37p (both have the variable 'p')
• 9.14q and -0.88q (both have the variable 'q')

Combining these terms means adding or subtracting their coefficients (the numerical part) and keeping the common variable part. This step simplifies the expression, making it easier to understand or use further.

Coefficients

Coefficients are the numerical parts of terms that are multiplied by the variable. For example, in 6.8p, the number 6.8 is the coefficient, and in -4.37p, the number -4.37 is the coefficient.
Identifying and combining coefficients is crucial when simplifying expressions. When we combine like terms, we specifically add or subtract their coefficients.
In the given exercise:

• We combined 6.8 and -4.37 for the terms with 'p'
• We combined 9.14 and -0.88 for the terms with 'q'

This combining process results in new coefficients for the simplified version of the expression: 2.43p + 8.26q.

Variable Terms

Variable terms are parts of an expression that contain a variable (like 'p' or 'q' in the exercise). They have two main parts: the coefficient and the variable component.
When simplifying, it is important to recognize and group these variable terms correctly. In the exercise, we saw variable terms with 'p' and 'q'.
Steps to simplify variable terms:

• Group terms with the same variable together
• Combine their coefficients
• Write the simplified expression with the common variable

Thus, correctly handling variable terms ensures an accurate simplification of algebraic expressions.