Problem 47 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(4(2 y+1)-6 y=18\)

Step-by-Step Solution

Answer

The solution to the given equation \(4(2 y+1)-6 y=18\) is \(y = 7\).

Step 1: Distribute the multiplication across parentheses

Multiply 4 by each term inside the parentheses \(2y + 1\), resulting in \(8y + 4 - 6y = 18\).

Step 2: Collect like terms

Combine like terms on the left side of the equation \(8y - 6y + 4 = 18\) that simplifies to \(2y + 4 = 18\).

Step 3: Subtract 4 from both sides

Subtraction of 4 from both sides of the equation results in \(2y = 14\).

Step 4: Solve for y

Divide by 2 on both sides \(\frac{2y}{2} = \frac{14}{2}\) simplifies to \(y = 7\).

Key Concepts

Distributive Property

When solving linear equations, you often encounter parentheses. To simplify expressions with parentheses, we use the distributive property. This property allows us to multiply a single term by each term inside a set of parentheses. For instance, in the equation \( 4(2y + 1) - 6y = 18 \), the number 4 is multiplied by both \( 2y \) and \( 1 \).

This results in creating two separate expressions: \( 4 \times 2y \) and \( 4 \times 1 \). Thus, you get \( 8y + 4 \). This step eliminates the parentheses and helps in breaking down complex equations into simpler ones.

• The distributive property is expressed as \( a(b + c) = ab + ac \).
• It makes complex problems manageable by simplifying terms.

Understanding this property is crucial because it is commonly used in algebraic manipulations.

Like Terms

Once you have used the distributive property, the next step is to look for like terms. Like terms are terms that have the same variable components raised to the same power. They are combined by adding or subtracting their coefficients.

In our example, after using the distributive property, we have \( 8y + 4 - 6y = 18 \). Here, \( 8y \) and \( -6y \) are like terms. By combining them, we simplify the equation to \( 2y + 4 = 18 \).

• Like terms must have identical variable parts.
• Combining them simplifies the equation further.

Recognizing and combining like terms effectively reduce the number of terms, making the equation easier to solve.

Solving Equations

With the like terms combined, we are closer to finding the solution. Solving equations involves isolating the variable on one side of the equation. This often involves a series of arithmetic operations to simplify the equation further.

In the current problem, once we have the simplified equation \( 2y + 4 = 18 \), we aim to isolate \( y \). We start by subtracting 4 from both sides, giving us \( 2y = 14 \).

• Operations performed on one side of the equation must be performed on the other side to maintain equality.
• Aim to have the variable term alone on one side.

The last step is dividing both sides by 2, which simplifies to \( y = 7 \). By performing these steps, you successfully find the value for \( y \), solving the initial equation.