Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the variable. \((11 a+3)-18 a+4\) for \(a=-2\)

Step-by-Step Solution

Answer

The result is 21.

Step 1: Identify and Simplify the Expression

The given expression is \((11a + 3) - 18a + 4\). First, combine like terms. Combine \(11a\) and \(-18a\) to get \(-7a\), and the constants \(3\) and \(4\) to get \(7\). This simplifies the expression to \(-7a + 7\).

Step 2: Substitute the Value of a

Substitute \(a = -2\) into the simplified expression \(-7a + 7\). This gives us \(-7(-2) + 7\).

Step 3: Perform the Calculation

Calculate \(-7(-2)\) which is \(14\), since multiplying two negative numbers results in a positive number. Then add \(7\) to \(14\), which results in \(21\).

Key Concepts

Simplifying Expressions

When tackling an algebraic expression, the first step is often to simplify it. Simplifying an expression means making it easier to understand and work with. This usually involves a few key actions:

• Removing any grouping symbols like parentheses.
• Combining like terms, which are terms with the same variables raised to the same power. This step not only reduces the number of terms but also simplifies calculations later on.
• Streamlining constants, which involves adding or subtracting the constant numbers together.

For example, in the original expression \((11a + 3) - 18a + 4\), we simplify by recognizing terms involving 'a' and the standalone numbers (constants). We combine \(11a\) with \(-18a\) to get \(-7a\). Similarly, the constants 3 and 4 are added together to make 7. The expression now reads \(-7a + 7\), a much neater form to work with.

Substitution Method

The substitution method is a powerful algebraic tool used to solve equations, especially when dealing with expressions that involve variables. It involves replacing a variable with a given value and calculating the result. Here's how we do it:

• Identify the variable you need to substitute.
• Substitute the given value for the variable throughout the expression.
• Solve the expression after substitution.

In our exercise, after simplifying the expression to \(-7a + 7\), we substitute \(a = -2\) into the equation. This means wherever you see 'a', you replace it with \(-2\). Our expression becomes \(-7(-2) + 7\). Replacing the variable directly allows us to calculate an exact numeric result.

Combining Like Terms

Combining like terms is a basic but crucial technique in algebra that involves merging terms that share the same variable, increasing efficiency in solving equations. This technique helps reduce the number of terms and simplifies expressions. Here’s a breakdown of how it works:

• Identify terms in the expression with the same variable (and the same powers if applicable).
• Combine these terms by adding or subtracting the coefficients (numbers in front of the variables).

In the example \((11a + 3) - 18a + 4\), the terms \(11a\) and \(-18a\) are alike as they both contain 'a'. By combining them, we subtract 18 from 11, resulting in \(-7a\). Doing this reduces the equation's complexity, turning it into \(-7a + 7\). This simplification is vital for both understanding and solving algebraic equations efficiently.