Problem 15 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(-4(x+6)=12\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = -9\).

Step 1: Distributing

Distribute '–4' to both 'x' and '6' on the left side of the equation to get -4x -24.

Step 2: Equating and Simplifying

So, the equation is -4x - 24 = 12. Now, add 24 to both sides to get -4x = 36.

Step 3: Solving for x

Finally, divide by -4 on both sides, to get the value of 'x' as -9.

Key Concepts

Equation Solving

Solving linear equations is a fundamental skill in algebra, essential for a strong understanding of mathematics. An equation is a statement that asserts the equality of two expressions, typically involving one or more variables. When solving equations, our aim is to find the value of the unknown variable that makes the equation true.

For instance, in the exercise \( -4(x+6)=12 \), we are tasked with finding the value of \( x \). The process involves several steps, each designed to simplify the equation step by step until \( x \)'s value is isolated and can be clearly determined. It's crucial to perform equivalent operations on both sides of the equation to maintain its balance, as changing only one side would render the equation untrue.

Distribution Property

The distribution property, often called the distributive law of multiplication over addition or subtraction, is a key concept for simplifying and solving equations. This property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.

For example, when we distribute \( -4 \) over \( x+6 \) in our exercise, we apply multiplication to each term inside the parenthesis: \( -4 \times x + (-4) \times 6 \), which simplifies to \( -4x - 24 \). Understanding this property is crucial, as it allows us to eliminate parentheses and combine like terms, which paves the way towards finding the solution.

Inverse Operations

Inverse operations are paired operations that undo each other—such as addition and subtraction or multiplication and division. These operations are the backbone of solving equations because they help us isolate the variable we're solving for.

In our given exercise, once the distribution is completed, we are left with \( -4x - 24 = 12 \). To isolate \( x \) we first use the inverse of subtraction, which is addition, to both sides to get rid of \( -24 \) from the left side. This leads us to \( -4x = 36 \). Next, to cancel out the multiplication by \( -4 \), we apply its inverse operation, division. Dividing both sides by \( -4 \) gives us the value of \( x \), which is \( -9 \). Familiarity with inverse operations makes equation solving a systematic and straightforward process.