Problem 85 - Algebra 2

Question

Evaluate each expression for the given values of the variable. \(4-2 k ; k=3,4,5,6,7\)

Step-by-Step Solution

Answer

For \(k=3, 4, 5, 6, 7\), the values are \(-2, -4, -6, -8, -10\) respectively.

Step 1: Substitute the First Value

Start with the expression \(4 - 2k\). Substitute \(k = 3\): \[ 4 - 2(3) = 4 - 6 = -2 \] So, when \(k=3\), the expression evaluates to \(-2\).

Step 2: Substitute the Second Value

Substitute \(k = 4\) into the expression:\[ 4 - 2(4) = 4 - 8 = -4 \] Thus, for \(k=4\), the expression evaluates to \(-4\).

Step 3: Substitute the Third Value

Substitute \(k = 5\):\[ 4 - 2(5) = 4 - 10 = -6 \] For \(k=5\), the evaluated expression is \(-6\).

Step 4: Substitute the Fourth Value

Substitute \(k = 6\):\[ 4 - 2(6) = 4 - 12 = -8 \] So, when \(k=6\), the expression equals \(-8\).

Step 5: Substitute the Fifth Value

Substitute \(k = 7\):\[ 4 - 2(7) = 4 - 14 = -10 \] For \(k=7\), the expression evaluates to \(-10\).

Key Concepts

Understanding the Substitution Method

The substitution method is a straightforward approach used to evaluate algebraic expressions. In simple terms, you replace a variable in the algebraic expression with a given number or value. This allows you to compute the expression's result with known numbers instead of unknown variables.
For example, consider the expression \(4 - 2k\). To evaluate it using the substitution method, you need specific values for \(k\). Each value is substituted into the expression to compute a numerical answer:

• First, if \(k = 3\), substitute it to get: \(4 - 2(3) = 4 - 6 = -2\).
• For \(k = 4\), substitute and solve: \(4 - 2(4) = 4 - 8 = -4\).

Understanding this method is crucial. It lays the groundwork for solving more complex mathematical problems by simplifying expressions using specific values.

Breaking Down Algebraic Expressions

An algebraic expression is a mathematical phrase combining numbers, variables, and sometimes operations like addition, subtraction, multiplication, or division. Expressions can take different forms, from very simple to quite complex, depending on the number of terms involved.
The expression \(4 - 2k\) comprises:

• A constant term \(4\), which is fixed and does not change.
• A variable term \(2k\), which varies depending on the value of \(k\). The coefficient "2" in \(2k\) indicates that \(k\) is doubled in the expression.
• Operators, specifically subtraction in this case, dictate how the terms interact.

By substituting different values for \(k\), you can evaluate how the expression behaves. This highlights the biggest benefit of algebraic expressions: flexibility.

Mastering Integer Operations

Integer operations are basic math actions performed on whole numbers, either positive or negative. They include addition, subtraction, multiplication, and division, all fundamental for manipulating algebraic expressions.
Let's consider the expression \(4 - 2k\):

• Subtraction occurs when evaluating \(4 - 6\); here, integers \(4\) and \(6\) work within this operation.
• Multiplication happens with \(2(3)\); the number \(3\) multiplies by the coefficient \(2\).

Understanding integer operations ensures you can confidently deal with positive and negative numbers, like moving from \(4 - 6 = -2\). Mastery of these operations is key to success in algebra and mathematics overall.