Problem 2 - Algebra A Combined Function

Question

Solve each equation. See Examples 1 and \(2 .\) $$ -3 x+1=-2(4 x+2) $$

Step-by-Step Solution

Answer

The solution is \(x = -1\).

Step 1: Distribute on the Right Side

First, distribute the \(-2\) across the expression inside the parentheses on the right side of the equation. This means you multiply \(-2\) by both \(4x\) and \(2\). This results in \(-8x - 4\). Now the equation is: \(-3x + 1 = -8x - 4\).

Step 2: Move All Terms Involving x to One Side

To start combining terms, aim to move all terms involving \(x\) to the left side. Add \(-8x\) to both sides to achieve: \(-3x + 8x = -8x + 8x - 4\). This simplifies the equation to \(5x + 1 = -4\).

Step 3: Move Constant Terms to the Opposite Side

To isolate terms involving \(x\) on one side, subtract \(1\) from both sides of the equation to get: \(5x = -4 - 1\). This further simplifies to \(5x = -5\).

Step 4: Solve for x

Finally, solve for \(x\) by dividing both sides by 5. This gives \(x = -5/5\), simplifying to \(x = -1\).

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental principle used in algebra to simplify expressions. It involves distributing a number outside of parentheses across the terms inside. In simple terms, you multiply the number by each term within the parentheses separately.

For example, in the expression \(-2(4x+2)\), the distributive property allows us to multiply \(-2\) by \(4x\) and \(2\) individually:

• Multiply \(-2\) by \(4x\) to get \(-8x\).
• Then, multiply \(-2\) by \(2\) to get \(-4\).

This results in the expression becoming \(-8x - 4\) once distribution is complete.

Understanding this property is crucial because it forms the basis for simplifying and eventually solving equations. It helps in ensuring that equations are easily manageable by breaking down expressions into simpler terms.

Techniques for Solving Linear Equations

Solving linear equations involves finding the value of the unknown variable that makes the equation true. The process requires rearranging the equation and simplifying it step by step.

Key steps include:

• Identifying terms involving the variable and constant terms.
• Manipulating the equation to position variable terms on one side.
• Moving constant terms to the opposite side.

For instance, in the equation \(-3x + 1 = -8x - 4\), initial steps include repositioning terms to a single side. We add \(-8x\) to both sides, turning the equation into \(5x + 1 = -4\).

This makes it easier to isolate the variable by performing operations like subtraction or division, eventually finding \(x = -1\). Attention to detail at each stage ensures accuracy and leads to the correct solution.

The Importance of Combining Like Terms

Combining like terms is a way to simplify expressions and make solving equations more straightforward. Like terms are terms that contain the same variable raised to the same power. It's an essential step in the simplification process.

In our example equation, \(-3x + 1 = -8x - 4\), we needed to rearrange the equation to bring all terms containing \(x\) together. By doing this, we combined \(-3x\) and \(8x\) (obtained from adding \(-8x\) to each side), yielding \(5x\).

• Combining terms helps reduce the complexity of an equation.
• It creates a clear path towards isolating the variable.

This concept is particularly important because it switches a potentially confusing equation into a manageable form, allowing for more efficient solutions. Understanding how and when to combine like terms can greatly enhance your problem-solving skills in algebra.