Problem 103 - Beginning and Intermediate Algebra

Question

If \(x=-5\) and \(y=-3,\) evaluate each expression. $$ 3 x+2 y $$

Step-by-Step Solution

Answer

The value of the expression is -21.

Step 1: Substitute Given Values into the Expression

Substitute the given values of \(x=-5\) and \(y=-3\) into the expression \(3x + 2y\). This will give us: \(3(-5) + 2(-3)\).

Step 2: Simplify the Multi-step Terms

First, simplify the multiplication of constants with the substituted values: \(3(-5) = -15\) and \(2(-3) = -6\).

Step 3: Perform Addition of Results

Now add the results from the simplified terms: \(-15 + (-6) = -21\).

Key Concepts

Substitution Method

The substitution method is a helpful approach used to simplify algebraic expressions by replacing variables with given numerical values. This method can be particularly useful when you have an expression with known values for variables and need to find the final numerical result.
Let's delve into the details and see how it applies to our exercise.

• Identify the variables in the expression you want to substitute. Typically, these are the unknowns, like \(x\) and \(y\) in our example.
• Carefully insert the provided values into the expression where the variables are located.

Once you've substituted the values, your expression should no longer contain any variables, only numbers. This allows you to tackle further steps like simplification and calculation.
Returning to our example, the expression \(3x + 2y\) becomes \(3(-5) + 2(-3)\) using substitution. This direct replacement is a simple yet powerful tool to evaluate algebraic expressions.

Simplification

Simplification in algebra involves breaking down expressions into more manageable numbers and making calculations easier. When you substitute variables with numbers in an expression, the next step is simplification.
For our example, after substitution, you obtain \(3(-5) + 2(-3)\):- Each part of the expression multiplies numbers (\(3(-5)\) and \(2(-3)\)).- First, calculate these products: \(3(-5) = -15\) and \(2(-3) = -6\).- With these simplified forms, addition comes next: \(-15 + (-6)\).Simplifying expressions like this makes complicated math problems simpler. It allows you to see the final outcome of an expression in a straightforward manner.

Integer Operations

Integer operations form the basis of many algebraic tasks, involving operations using whole numbers which can be positive or negative. Let's explore them further in the context of our exercise.

• Addition and Subtraction: When adding integers, especially negative ones, pay attention to signs. Adding a negative integer is equivalent to subtraction. For example, \(-15 + (-6)\) is like subtracting \(6\) from \(-15\).
• Multiplication and Division: Multiplying two negative integers results in a positive product, while multiplying a positive with a negative integer results in a negative product.

In our exercise, the integer operations are crucial; you perform both multiplication and addition. When you correctly handle signs, like multiplying \(3\) and \(-5\) or summing \(-15\) and \(-6\), it ensures accuracy. Understanding these concepts is essential when evaluating and simplifying algebraic expressions containing integers.