Problem 65 - Prealgebra

Question

Evaluate each expression if \(a=5, b=1,\) and \(c=3\) $$c a-c b$$

Step-by-Step Solution

Answer

The value of the expression is 12.

Step 1: Substitute Values

Replace the variables in the expression with the given values. For the expression \(c a - c b\), substitute \(a = 5\), \(b = 1\), and \(c = 3\).

Step 2: Perform Multiplication

Calculate the products for each term separately. First, compute \(c a = 3 \times 5 = 15\). Next, compute \(c b = 3 \times 1 = 3\).

Step 3: Subtraction

Subtract the second product from the first to complete the evaluation: \(15 - 3 = 12\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are like sentences in mathematics. They are formed by combining numbers, operators, and variables together. In an expression like \(c a - c b\), the letters \(a, b, \) and \(c\) are known as variables. Variables represent unknown values that can change. Think of them as placeholders that will hold numbers when we want to evaluate the expression.
Use algebraic expressions to model real-world situations and solve problems involving various unknowns. They make complex calculations organized and simpler.

• Terms: Each part of an expression separated by a plus or minus sign is called a term. In our expression, \(c a\) and \(c b\) are the terms.
• Operators: These are the plus, minus, multiplication, and division signs like in our expression, where the minus sign (-) is separating the two terms. This gives us direction on which operations to carry out.
• Coefficients: The numbers that are multiplied by the variables; in our example, 3 is a coefficient of both terms before substituting.

The Process of Variable Substitution

Variable substitution is a fundamental step in solving algebraic expressions. It's the process where each variable in the expression is replaced by a given number. This step simplifies the expression, turning it into a simple arithmetic operation.
In the example \(c a - c b\), you substitute \(a = 5\), \(b = 1\), and \(c = 3\), which are provided values. The expression becomes easier to handle: replace each instance of the variable with its given value.

• Find the values of each variable. It helps to note them down for clarity.
• Replace the variables in the expression with these specific numbers.
• Simplify further using basic arithmetic operations.

This straightforward process allows solving the expression step-by-step, ensuring accuracy and simplicity.

Performing Basic Arithmetic Operations

Once variables are substituted with numbers, you use basic arithmetic operations to evaluate the expression. You may already know some of these operations: addition, subtraction, multiplication, and division.
In this particular problem, first perform the multiplication: each term independently.

• Multiply \(c\) and \(a\): \(3 \times 5 = 15\).
• Multiply \(c\) and \(b\): \(3 \times 1 = 3\).

After completing the multiplications, focus on combining the results by performing subtraction:

• Subtraction: Take the result from the first multiplication and subtract the second: \(15 - 3 = 12\).

These basic operations form the building blocks of math and are useful in more complex calculations as well.