Problem 21 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ -5 s-2 t+1, \text { if } s=2 \text { and } t=-2 $$

Step-by-Step Solution

Answer

The value of the expression is -5.

Step 1: Substitute the known values

Substitute the given values of \(s\) and \(t\) into the expression. Here, \(s = 2\) and \(t = -2\). The expression becomes:\[-5(2) - 2(-2) + 1\]

Step 2: Perform multiplication

Calculate the results of the multiplications:\[-5 \times 2 = -10\] \[-2 \times (-2) = 4\]So, the expression is now:\[-10 + 4 + 1\]

Step 3: Perform addition

Now, add the results of the previous calculations:\[-10 + 4 = -6\] Then add 1:\[-6 + 1 = -5\]

Step 4: Verify the calculation

Recheck each arithmetic operation to ensure there are no errors:- Multiplication: \(-5 \times 2 = -10\) and \(-2 \times (-2) = 4\).- Addition: \(-10 + 4 = -6\), and \(-6 + 1 = -5\).The calculations are confirmed to be correct.

Key Concepts

Substitution in Algebra

Substitution in algebra is an essential technique used to simplify expressions by replacing variables with their actual values. This process often makes evaluating expressions much easier. Here's how you can do it step by step:

• Identify the variables in the expression you are dealing with. In our example expression, we have variables \(s\) and \(t\).
• Replace each variable with its given value. This is the essence of substitution. For instance, in the expression \(-5s - 2t + 1\), we substitute \(s = 2\) and \(t = -2\), transforming it into \(-5(2) - 2(-2) + 1\).
• After substitution, proceed with arithmetic operations to simplify the expression.

This method is particularly helpful for solving equations and checking solutions. Practice substitution with different expressions to get comfortable with the process.

Multiplying Integers

Multiplying integers is a fundamental arithmetic operation that involves finding the product of two whole numbers. It is important to remember the sign rules when multiplying integers. Here are some key points to consider:

• Multiplying two positive numbers always gives a positive product. For example, \( 3 \times 4 = 12 \).
• The product of two negative numbers is also positive. For instance, \(-3 \times -4 = 12\).
• However, if you multiply a positive number by a negative number, the product is negative. For example, \(5 \times -2 = -10\).

In the exercise, we multiplied integers as part of the expression evaluation: \(-5 \times 2\) resulted in \(-10\), and \(-2 \times -2\) resulted in \(+4\). Remember these rules to help simplify your work with integers.

Adding Integers

Adding integers requires an understanding of how positive and negative numbers interact on the number line. The basics of adding integers include:

• When adding two positive integers, the result is positive, such as \(2 + 3 = 5\).
• Adding two negative integers results in a negative sum, for example, \(-2 + -3 = -5\).
• If you are adding a positive integer to a negative integer, consider their absolute values. For instance, \(-6 + 4\): Subtract the smaller absolute value from the larger and take the sign of the number with the larger absolute value, resulting in \(-2\).

In the original expression \(-10 + 4 + 1\), we added integers step-by-step. Adding \(-10 + 4\) gives \(-6\), then adding \(-6 + 1\) gives \(-5\). Understanding these principles is vital for combining integers correctly in various mathematical contexts.