Problem 55 - Prealgebra

Question

Evaluate each expression if \(a=6, b=4,\) and \(c=5\). $$a+c-b$$

Step-by-Step Solution

Answer

The value of the expression is 7.

Step 1: Substitute Values

Replace each variable in the expression with the given values: replace \(a\) with 6, \(c\) with 5, and \(b\) with 4. The expression becomes \(6 + 5 - 4\).

Step 2: Add the First Two Terms

Add the first two terms in the expression: \(6 + 5 = 11\), so the expression now is \(11 - 4\).

Step 3: Subtract the Last Term

Subtract the final term from the result obtained: \(11 - 4\), which simplifies to \(7\).

Key Concepts

Substitution in Algebra

Substitution is a fundamental concept in algebra that involves replacing variables with their respective values. It is an essential skill for solving algebraic expressions, as it allows us to simplify and evaluate them. To substitute effectively:

• Identify the variables in the expression.
• Replace each variable with the number it represents. For example, if \(a = 6\), substitute \(6\) where \(a\) appears in the expression.
• Ensure all variable replacements are done correctly to avoid errors.

Practicing substitution helps to build a solid foundation for more advanced algebraic problems. It also aids in understanding how expressions change when different values are applied.

Order of Operations

Order of operations is a critical concept that dictates the sequence in which arithmetic operations should be performed to achieve consistent results. The standard sequence is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Here’s how it applies:

• Always perform operations inside parentheses first.
• Handle any exponents (powers) next.
• Proceed with multiplication and division, moving left to right.
• Finally, carry out addition and subtraction, also from left to right.

By following these rules, you ensure that expressions are solved systematically, avoiding mistakes in computation. In our exercise, we started with addition before subtraction because there were no parentheses, multiplication, or division involved.

Basic Arithmetic Skills

Basic arithmetic forms the foundation for dealing with numbers and mathematical operations. It includes addition, subtraction, multiplication, and division. These skills are not only essential for everyday calculations but are also integral to tackling more complex mathematical problems. Here's a quick reminder:

• Addition: Combines numbers to get a sum.
• Subtraction: Determines the difference between numbers.
• Make sure these operations are performed accurately to avoid errors in results.

These operations are directly applicable in our current exercise. We added 6 and 5, resulting in 11, and then subtracted 4 to eventually get 7. Honing these skills will make algebra and other areas of mathematics much easier to handle.

Prealgebra Concepts

Prealgebra lays the groundwork needed for mastering algebra by introducing basic mathematical principles. It typically covers topics such as number operations, the use of variables, solving simple equations, and understanding expressions. Key aspects include:

• Understanding Variables: Symbols that hold a place for numbers in expressions.
• Equations: Mathematical statements showing the equality of two expressions.
• Simple Problem Solving: Techniques to find unknown values in mathematical problems.

Grasping these concepts can make the transition to algebra smoother and less intimidating. In our exercise, we used variables and replaced them with numbers, a common task in prealgebra that paves the way for understanding more complex equations later.