Problem 37 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 4(2 x+3)=-4 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -2.\)

Step 1: Distribute

Distribute the 4 to the terms inside the parenthesis: \(4*2x + 4*3= -4\), which simplifies to \(8x + 12 = -4.\)

Step 2: Isolate x

To isolate x, subtract 12 from both sides of the equation: \(8x + 12 -12 = -4 -12,\) which becomes \(8x = -16.\)

Step 3: Solve for x

To find the value of x, divide both sides of the equation by 8: \(8x /8 = -16 /8,\) and this simplifies to \(x = -2.\)

Key Concepts

Understanding the Distributive Property

The distributive property is a useful algebraic tool that helps us simplify expressions and solve equations. It is especially handy when dealing with parentheses. This property allows us to distribute, or spread out, multiplication over addition or subtraction within parentheses.
For example, in the given problem, we have the expression \(4(2x + 3)\). The distributive property tells us we can multiply 4 by each term inside the parentheses:

• First, multiply 4 by \(2x\), giving us \(8x\).
• Next, multiply 4 by 3, resulting in 12.

By distributing the 4, we transform \(4(2x + 3)\) into \(8x + 12\). This is a key first step before moving on to solve the equation. By understanding how to distribute terms, you can break down complex expressions into manageable parts.

Mastering the Isolation of Variables

In any equation, isolating the variable is a crucial step to finding its value. To "isolate" means to get the variable by itself on one side of the equation.
In the exercise, after distributing, we are left with \(8x + 12 = -4\). Our goal is to isolate \(x\). To do this, we need to remove the 12 that is added to \(8x\).

• Subtract 12 from both sides of the equation, which keeps the equation balanced.
• This gives us \(8x + 12 - 12 = -4 - 12\).
• Simplifying both sides results in \(8x = -16\).

Now the \(x\) term is by itself on one side, with its coefficient (8) adjoining it. Remember, isolating the variable is like peeling an onion - it often involves removing layers (terms) step by step.

Essentials of Solving Equations

Once we have isolated the variable, the final step is to solve for its value. This involves performing operations that will give the variable its own identity - literally finding what \(x\) equals.
In our example, with the isolated equation \(8x = -16\), we solve for \(x\) by removing the coefficient 8 that is currently multiplying \(x\).

• To do this, divide both sides of the equation by 8, which results in \(x\) standing alone.
• This calculation is \(8x / 8 = -16 / 8\).
• Once simplified, it gives us \(x = -2\).

This procedure of solving for the variable by getting it alone is common in algebraic equations. Remember: whatever you do on one side of the equation, do the same on the other to maintain balance. This is the essence of solving equations.