Problem 58 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(-3.4 p-0.8+2.5+7.2 p\)

Step-by-Step Solution

Answer

The simplified expression is 3.8p + 1.7.

Step 1 - Separate terms involving the variable

Identify and group together the terms that involve the variable, which in this case is 'p'. The terms involving 'p' are (-3.4p) and (7.2p).

Step 2 - Separate the constant terms

Next, identify and group the constant terms (terms without the variable 'p'). These terms are (-0.8) and (2.5).

Step 3 - Combine the variable terms

Add the coefficients of the terms involving 'p'. Compute (-3.4 + 7.2) to simplify the variable terms. This results in 3.8p.

Step 4 - Combine the constant terms

Add the constant terms. Compute (-0.8 + 2.5) which results in 1.7.

Step 5 - Write the simplified expression

Combine the simplified variable and constant terms to express the final simplified form. Therefore, the simplified expression is 3.8p + 1.7.

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one of the primary steps is combining like terms. Like terms are terms in an expression that have the same variable raised to the same power. For example, in the expression \(-3.4p - 0.8 + 2.5 + 7.2p\), the like terms are \(-3.4p\) and \(7.2p\).Combining like terms means adding or subtracting their coefficients while keeping the variable part the same. This helps to consolidate the expression into a simpler form. In our example, the coefficients of \(p\) are \(-3.4\) and \(7.2\). When combined, these become \(3.8p\). The constant terms (those without a variable) are combined in a similar way to further simplify the expression.

Coefficients

A coefficient is the numerical part of a term that contains a variable. In other words, it’s the number that is multiplied by the variable in a term.For example, in the term \(-3.4p\), \(-3.4\) is the coefficient, and in \(7.2p\), \(7.2\) is the coefficient.When you combine like terms, you add or subtract the coefficients of the variable terms. This allows you to simplify the variable part of the expression. In our exercise, combining \(-3.4p\) and \(7.2p\) involves adding the coefficients \(-3.4\) and \(7.2\), resulting in \(3.8p\).

Constants

Constants are terms in an expression that do not contain variables. They are simply numbers. In the expression \(-3.4p - 0.8 + 2.5 + 7.2p\), the constants are \(-0.8\) and \(2.5\).When simplifying an expression, constants are combined together just like like terms with variables. You add or subtract them directly. In our example, combining the constants \(-0.8\) and \(2.5\) gives us \(1.7\). This results in the simplified constant part of the expression.

Variable Terms

Variable terms are parts of an expression that include a variable along with a coefficient. These terms can be added or subtracted from one another if they have the same variable and exponent. In our example, the variable terms are \(-3.4p\) and \(7.2p\).Each of these terms contains the variable \(p\), so they can be combined directly by their coefficients. After combining, the simplified variable term is \(3.8p\). This makes the overall expression more manageable and easier to work with.