Problem 3 - Elementary Algebra

Question

Evaluate for the given set of \(x\) -values. $$ 1 x+9 ;\\{-10,-9,0\\} $$

Step-by-Step Solution

Answer

The results are -1 for \(x = -10\), 0 for \(x = -9\), and 9 for \(x = 0\).

Step 1: Understanding the Problem

The expression given is \(1x+9\), which means we need to evaluate this expression for each \(x\)-value in the set \(\{-10, -9, 0\}\). We'll substitute each \(x\)-value into the expression and calculate the result.

Step 1 - Evaluate for \(x = -10\)

Substitute \(-10\) into the expression: \[ 1(-10) + 9 = -10 + 9 = -1. \] So, the result when \(x = -10\) is \(-1\).

Step 2 - Evaluate for \(x = -9\)

Substitute \(-9\) into the expression: \[ 1(-9) + 9 = -9 + 9 = 0. \] Thus, when \(x = -9\), the result is \(0\).

Step 3 - Evaluate for \(x = 0\)

Substitute \(0\) into the expression: \[ 1(0) + 9 = 0 + 9 = 9. \] Thus, when \(x = 0\), the result is \(9\).

Key Concepts

Substitution Method

The substitution method is a cornerstone concept in mathematics, particularly when dealing with expressions. It involves replacing a specific variable with a given value to simplify or evaluate an expression. In our example, the expression is given as \(1x+9\). Instead of solving for \(x\), we substitute various values of \(x\) (from a given set) into the expression to find the outcome. This method is simple to use:

• Select the given value from the set.
• Replace the variable \(x\) in the expression with this value.
• Perform arithmetic operations to simplify the expression.

For instance, when substituting \(x = -10\), you conduct the operation as \(1(-10) + 9\). With careful arithmetic, you resolve it to find the result for that specific \(x\), which in our case, was \(-1\). This method helps efficiently check or find multiple outcomes using a single algebraic expression.

Algebraic Expressions

Understanding algebraic expressions is fundamental to mastering algebra. An algebraic expression is a mathematical phrase that combines numbers, variables, and operation symbols. Variables like \(x\) can represent unknown numbers or can change depending on the context in which they appear. Our expression here is \(1x + 9\), which has:

• "1x", meaning \(1\) multiplied by \(x\).
• "9", a constant term that is being added.

The beauty of an algebraic expression is its ability to represent real-world scenarios with variables that can change. This allows us to manipulate these expressions by following rules and laws of arithmetic and algebra, such as commutative, associative, or distributive laws, to simplify or solve them. By understanding what each component of an expression stands for, you're better equipped to evaluate it for a given set of values.

Set of Values

A set of values is a collection of numbers that you can use for different processes in algebra. When evaluating expressions, a set helps determine how the expression behaves with different inputs. In our example, the set \(\{-10, -9, 0\}\) is used to evaluate the expression \(1x + 9\). Each element in the set replaces the variable in the expression, providing distinct outcomes.

• This practice shows how an expression changes as \(x\) shifts values.
• It creates a clearer panorama of how inputs affect outputs.
• Helps in finding possible values and scenarios in real-world applications.
• Useful in understanding patterns and seeing practical applications like trends or predictions.

Employing sets in expressions lets you explore various outcomes without altering the structure of the equation. This fosters a deeper comprehension of the role variables play in algebra.