Problem 25 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ -x+2 y, \text { if } x=-7 \text { and } y=-1 $$

Step-by-Step Solution

Answer

The value of the expression is 5.

Step 1: Substitute Values into Expression

Given the expression \(-x + 2y\), we need to substitute the given values for \(x\) and \(y\). Replace \(x\) with \(-7\) and \(y\) with \(-1\). Thus the expression becomes: \(-(-7) + 2(-1)\).

Step 2: Simplify the Expression

Simplify each part of the expression separately. First, consider \(-(-7)\), which simplifies to \(+7\) since the negative sign in front of \(-7\) changes the sign. Then, calculate \(2(-1)\), which equals \(-2\).

Step 3: Combine the Results

Now that we have simplified parts of the expression, combine them: \(7 + (-2)\). This simplifies to \(7 - 2 = 5\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that allows us to make expressions easier to understand and solve. When you simplify an expression, you are combining like terms and reducing the expression to its simplest form. This process often involves arithmetic operations and understanding how to appropriately apply the rules of algebra.Let's take a quick look at the initial expression we have: - Say, \(-x + 2y\).This expression consists of different variables, namely \(x\) and \(y\), both of which have coefficients. To simplify a given expression after substituting numbers in place of variables, follow these basic steps:

• Identify and substitute the values of variables from the information provided.
• Apply arithmetic operations like addition, subtraction, multiplication, and division properly.
• Combine like terms whenever possible.

In our example, after substituting \(x = -7\) and \(y = -1\), we dealt with the expression \(-(-7) + 2(-1)\). Each part was simplified: the negative of a negative gave us a positive, showcasing the rule that two negatives make a positive, and then multiplying 2 by -1 gave us -2. Finally, combining gives us \(7 - 2\), which is straightforward to complete.

Basic Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In algebra, letters are often used to represent numbers. The example discussed falls well within the domain of basic algebra where substitution skills are applied to solve expressions.The key points in understanding basic algebra include:

• Recognizing algebraic expressions, like \(-x + 2y\), which consists of terms and operations.
• Understanding the meaning of variables: \(x\) and \(y\) in an expression are placeholders for numbers.
• Learning how to substitute numbers for variables to evaluate expressions.
• Applying fundamental arithmetic operations in the order dictated by the expression.

These concepts help students manipulate equations and expressions to discover unknown values. Substituting the given numbers into the expression demonstrated how numbers can replace variables and simplify into a result using arithmetic rules.

Evaluating Expressions

Evaluating expressions is all about finding the value of the expression once the variables are replaced by actual numbers. This is a crucial step in solving mathematical problems and requires a precise approach to ensure the correct results.To evaluate an expression like \(-x + 2y\) when given specific values for \(x\) and \(y\):

• Substitute the given values into the expression: replace \(x\) with \(-7\) and \(y\) with \(-1\).
• Perform operations step by step, ensuring each component of the expression is calculated correctly. In this case, calculate \(-(-7)\) and \(2(-1)\).
• Simplify the results to find the final value. Here, it directs to \(7 - 2\), giving you 5.

By evaluating expressions correctly, one obtains a numeric answer which clarifies the problem at hand. It reaffirms the discipline in following mathematical rules to arrive at a logical conclusion, as showcased by our solution resulting in the value of 5.