Problem 29 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the variable. \(4 y+8-2 y\) for \(y=3\)

Step-by-Step Solution

Answer

The value of the expression is 14.

Step 1: Substitute the given value of y

Replace the variable \( y \) in the expression \( 4y + 8 - 2y \) with the given value \( y = 3 \). This gives us the expression: \( 4(3) + 8 - 2(3) \).

Step 2: Simplify the expression

First, perform the multiplication operations in the expression \( 4(3) + 8 - 2(3) \). Calculate \( 4 imes 3 = 12 \) and \( 2 imes 3 = 6 \), resulting in the expression: \( 12 + 8 - 6 \).

Step 3: Perform addition and subtraction

Now, perform the addition \( 12 + 8 = 20 \), followed by the subtraction \( 20 - 6 = 14 \). Thus, the simplified value of the expression is \( 14 \).

Key Concepts

Solving Equations

Solving equations involves finding the value of a variable that makes an equation true. In many exercises, you're asked to find the unknown variable that satisfies the given equation. This process is fundamental to understanding how relationships between numbers work.

To solve an equation, you typically isolate the variable on one side of the equation while keeping the equation balanced. Think of it like maintaining balance in a seesaw; you want to keep both sides equal.

Here are some key points to remember when solving equations:

• Always perform the same operation on both sides of the equation.
• Combine like terms before isolating the variable.
• Use inverse operations to cancel out numbers around the variable.

In our exercise, solving the equation essentially means finding the result after substituting the value into the expression and performing the necessary arithmetic operations.

Variable Substitution

Variable substitution is when you replace a variable in an expression with a given number. This makes it possible to evaluate an expression and find a numerical result. In our example, we substituted the variable \( y \) with \( 3 \).

Here’s how variable substitution helps you:

• Simplifies complex expressions by turning variables into numbers.
• Makes it easier to see the impact of changing variables on the expression's outcome.

After substituting \( y = 3\) into the expression \( 4y + 8 - 2y \), the expression became \( 4(3) + 8 - 2(3)\). This step is crucial as it sets the stage for further simplification and calculation.

Simplification of Expressions

Simplification of expressions is the process of making an expression simpler and more manageable. This is done by combining like terms, performing arithmetic operations, and reducing the expression to its simplest form.

In our example, after substitution, the expression \( 4(3) + 8 - 2(3) \) was simplified by first performing multiplication, yielding \( 12 + 8 - 6 \).

Key tips for simplifying expressions:

• Always work with the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS).
• Combine like terms, which are terms that have the same variable part.
• Simplify step-by-step to avoid errors.

Once you have a simplified expression, it's easy to see the final result, which, in this case, comes out to be \( 14\). The simplification process turns a complex expression into a straightforward number.