Problem 45 - Intermediate Algebra

Question

Evaluate the algebraic expressions for the given values of the variables. $$ 7 a-2 b-9 a+3 b, \quad a=4 \text { and } b=-6 $$

Step-by-Step Solution

Answer

The value of the expression is \(-14\).

Step 1: Substitute the Values of Variables

Replace each 'a' with 4 and each 'b' with -6 in the expression \( 7a - 2b - 9a + 3b \). This gives us \( 7(4) - 2(-6) - 9(4) + 3(-6) \).

Step 2: Simplify the Expression

Calculate each term using the substituted values: \[ 7(4) = 28, \quad -2(-6) = 12, \quad -9(4) = -36, \quad 3(-6) = -18 \]. The expression is now: \( 28 + 12 - 36 - 18 \).

Step 3: Combine Like Terms

Add or subtract the terms in the expression step by step: 1. Combine \( 28 + 12 = 40 \).2. Combine \( -36 - 18 = -54 \).

Step 4: Final Result

Now subtract \( 54 \) from \( 40 \), giving us \( 40 - 54 = -14 \). Thus, the value of the expression is \(-14\).

Key Concepts

Variables Substitution

Variables substitution is an essential part of evaluating algebraic expressions. You replace the variables in the expression with given numeric values. This process allows us to convert algebraic expressions into a form that allows for direct calculation.

• For example, in the expression \(7a - 2b - 9a + 3b\), if we're given that \(a = 4\) and \(b = -6\), we "substitute" these numbers in place of \(a\) and \(b\).
• So, \(7a\) becomes \(7(4)\), \(-2b\) becomes \(-2(-6)\), \(-9a\) becomes \(-9(4)\), and \(3b\) becomes \(3(-6)\).

This replacement simplifies the expression into a series of arithmetic operations that are straightforward to manage.

Simplifying Expressions

Once the values have been substituted into an expression, the next step is to simplify it. Simplifying means calculating the numerical expressions to reduce the entire expression into a simpler form.

• Consider the expression after variables substitution: \(7(4) - 2(-6) - 9(4) + 3(-6)\).
• Simplify each of these terms: \(7(4) = 28\), \(-2(-6) = 12\), \(-9(4) = -36\), \(3(-6) = -18\).

The expression simplifies to \(28 + 12 - 36 - 18\). Each term is now a simple number instead of a combination of variables and coefficients, making it easier to work with.

Combining Like Terms

Combining like terms is a crucial step in simplifying expressions further. It involves adding or subtracting terms that have been simplified to similar forms. In arithmetic, this means dealing with numbers directly.

• In the expression \(28 + 12 - 36 - 18\), we combine the results: First, add \(28\) and \(12\) to get \(40\).
• Then, add \(-36\) and \(-18\) to get \(-54\).
• Finally, subtract the sum of negative terms from the sum of positive terms: \(40 - 54\).

The process of combining like terms helps in getting the final numeric results. In this case, it results in \(-14\), the evaluated value of the original expression.