Problem 36 - Algebra 2

Question

Evaluate each expression for \(a=3\) and \(b=-5\). \(a-b\)

Step-by-Step Solution

Answer

The value of the expression when a is 3 and b is -5 is 8.

Step 1: Substitute the values of a and b

Replace the variable 'a' with 3 and 'b' with -5 in the expression 'a - b'. So we substitute and get: 3 - (-5).

Step 2: Simplify the expression

Simplify the expression by changing the minus a negative to plus a positive. This is because subtracting a negative is the same as adding the positive of that number. Therefore, 3 - (-5) becomes 3 + 5.

Step 3: Perform the addition

Add the two numbers together. 3 + 5 equals 8.

Key Concepts

Substitution in Algebra

At the heart of understanding algebraic expressions lies the concept of substitution. This is where we take a given expression and replace the variables—letters used to represent unspecified numbers or values—with known values. In the context of our exercise, the variables are 'a' and 'b', and the known values are 3 and -5, respectively.

Implementing this process effectively requires careful attention to ensure that each variable is replaced with its corresponding value. For instance, when given the expression \(a - b\), and the values \(a = 3\) and \(b = -5\), we substitute the variables with these numbers, resulting in \(3 - (-5)\). It's essential to preserve the signs attached to each number during substitution, as this impacts the final result.

Simplifying Expressions

Simplification is a crucial step that follows substitution. It involves manipulating the expression to its simplest form, making it easier to understand and solve. In simplifying expressions, we use the rules of arithmetic to combine like terms and apply mathematical properties such as the distributive, associative, and commutative properties.

In our exercise, we simplify the expression after substitution by dealing with the double negative. A negative sign in front of a parenthesis indicates that we need to change the sign of the number inside. Therefore, when we see \(3 - (-5)\), we are really seeing two negatives, which become a positive. Hence, the expression simplifies to \(3 + 5\). Keeping these basic rules in mind helps simplify even the most complex expressions.

Arithmetic Operations in Algebra

Algebra is not just about working with variables; it also encompasses the use of arithmetic operations such as addition, subtraction, multiplication, and division. These operations are fundamental in algebra and follow the same principles as they do in basic arithmetic.

After simplifying the expression to \(3 + 5\), we encounter the final step: performing the arithmetic operation. In this case, it’s an addition problem. To obtain the solution, we just add the two numbers together which gives us the result of 8. It is essential to execute these operations correctly, as they form the basis of more advanced topics in algebra.