Problem 45 - Algebra A Combined Function

Question

Solve. $$ 7(6+w)=6(2+w) $$

Step-by-Step Solution

Answer

The solution to the equation is \(w = -30\).

Step 1: Distribute the Constants

Start by distributing the constants 7 and 6 in the respective expressions on both sides of the equation. For the left side, distribute 7: \[ 7(6 + w) = 7 \cdot 6 + 7 \cdot w = 42 + 7w \]For the right side, distribute 6:\[ 6(2 + w) = 6 \cdot 2 + 6 \cdot w = 12 + 6w \]

Step 2: Form the Equation

Equate the distributed expressions from both sides together:\[ 42 + 7w = 12 + 6w \]

Step 3: Isolate the Variable

Subtract \(6w\) from both sides to remove \(w\) from the right side of the equation:\[ 42 + 7w - 6w = 12 + 6w - 6w \]This simplifies to:\[ 42 + w = 12 \]

Step 4: Solve for the Variable

Subtract 42 from both sides to isolate \(w\):\[ 42 + w - 42 = 12 - 42 \]This simplifies to:\[ w = -30 \]

Key Concepts

Distributive Property

The distributive property is a helpful tool in algebra that simplifies expressions by distributing a single term across terms inside a parenthesis. In the original problem, you apply the distributive property to both sides of the equation.You start with:

• Left side: \[ 7(6 + w) \]
• Right side: \[ 6(2 + w) \]

To distribute, multiply each term inside the parentheses by the constant outside:

• For the left side: \[ 7 \times 6 + 7 \times w = 42 + 7w \]
• For the right side: \[ 6 \times 2 + 6 \times w = 12 + 6w \]

This step ensures that every term inside the parentheses is multiplied individually by the number outside. Remember: The distributive property helps break down equations into simpler, more manageable parts.

Isolating Variables

Isolating the variable is crucial when solving equations because it helps you find the value of the unknown. In the solved exercise, the ultimate goal is to isolate the variable \( w \) on one side of the equation.You first take:\[ 42 + 7w = 12 + 6w \]Then, subtract \( 6w \) from both sides to simplify:\[ 42 + 7w - 6w = 12 + 6w - 6w \]Which leads to:\[ 42 + w = 12 \]This step ensures \( w \) is on one side, making it easier to solve for.
Keeping the equation balanced is key; whatever you do to one side, do the same to the other to maintain equality.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the unknown variable's value. After isolating terms, the next step in solving the equation is straightforward.In the exercise:

• You're left with:\[ 42 + w = 12 \]
• Subtract \( 42 \) from both sides to solve for \( w \):\[ w = 12 - 42 \]
• This results in:\[ w = -30 \]

Solving equations involves logical reasoning, considering equation balance, and finding a practical approach to isolate and calculate values.

Algebraic Expressions

Algebraic expressions play a vital role in understanding linear equations. They combine numbers, variables, and arithmetic operations to represent a calculation or value.For example, in the exercise, you encounter:

• \[ 7(6 + w) \] and \[ 6(2 + w) \]

These are algebraic expressions where numbers (\( 6, 7, \) and \( 2 \) ) are multiplied by variables (\( w \) ).Learning how to manipulate these expressions using properties like the distributive property helps solve occasions smoothly. The transition from complex expressions to a simplified equation is the essence of understanding algebra. Simplifying expressions is a stepping stone to resolving equations effectively.