Problem 31 - Fundamentals of Mathematics

Question

Find the value of each expression. $$9 a+6 b-8 x+4 y, \text { if } a=-2, b=-1, x=-2, \text { and } y=0$$

Step-by-Step Solution

Answer

The value of the expression is -8.

Step 1: Substitute Variables

Replace the variables in the expression with the given values: \( a = -2 \), \( b = -1 \), \( x = -2 \), and \( y = 0 \). The expression becomes \( 9(-2) + 6(-1) - 8(-2) + 4(0) \).

Step 2: Calculate Each Term

Evaluate each term separately: \( 9(-2) = -18 \), \( 6(-1) = -6 \), \(-8(-2) = 16 \), and \( 4(0) = 0 \).

Step 3: Sum the Results

Add all the calculated values from Step 2: \( -18 + (-6) + 16 + 0 \).

Step 4: Final Calculation

Perform the final addition: \( -18 - 6 + 16 + 0 = -8 \).

Key Concepts

Substitution in Algebra

Substitution is a powerful tool in algebra that allows us to evaluate expressions by plugging in specific values for the variables. Think of it as replacing a placeholder with an actual number so that we can conduct mathematical operations. In the exercise given, we're tasked to find the value of an algebraic expression by substituting each variable with its assigned value.

• Identify the variables in your expression: These are usually represented by letters such as \( a, b, x, \) and \( y \).
• Know the specific values to substitute: For this problem, \( a = -2 \), \( b = -1 \), \( x = -2 \), and \( y = 0 \).
• Replace the variables: Directly plug the values into the expression wherever these variables appear, making it ready for evaluation.

This process simplifies an otherwise complex-looking expression into a more straightforward arithmetic operation that can be solved step-by-step.

Understanding Expressions

Expressions in algebra represent combinations of numbers, variables, and mathematical operations such as addition or subtraction. They do not equate to a fixed value unless the variables within them are given specific numbers to replace them, called substitution.

• An algebraic expression in this exercise is \( 9a + 6b - 8x + 4y \).
• Variables: These are symbols that can take on different values. In this expression, they are \( a, b, x, \), and \( y \).
• Coefficients: These are the numbers multiplying the variables, such as \( 9 \) for \( a \), \( 6 \) for \( b \), etc.

Expressions allow us to symbolically represent real-world problems in mathematics. By understanding how to manipulate and evaluate them, we can solve equations and find solutions to various problems.

Evaluation of Expressions

Evaluation is the process of finding the numerical value of an expression once the variables have been substituted with specific numbers. Following the steps of substitution helps set the stage for effective evaluation.

• Calculate each term: Once values are substituted, perform mathematical operations on each term individually: \( 9(-2) = -18 \), \( 6(-1) = -6 \), \(-8(-2) = 16 \), and \( 4(0) = 0 \).
• Sum the terms: Combine the results of your computations by adding or subtracting them: \( -18 + (-6) + 16 + 0 \).
• Reach the final value: Through careful addition or subtraction, arrive at the solution: \( -8 \).

Through evaluation, you break down the expression systematically to find a single, definitive answer. This step-by-step process increases clarity and reduces errors in algebraic problem-solving.