Problem 12 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$4(-2 x+1)=6-(2 x-4)$$

Step-by-Step Solution

Answer

x = -1

Step 1 - Distribute the constant

Distribute the 4 on the left side of the equation: $$ 4(-2x + 1) = 4 \times -2x + 4 \times 1 $$ This simplifies to: $$ -8x + 4 $$

Step 2 - Simplify the right side

Simplify the right side of the equation by distributing the minus sign: $$ 6 - (2x - 4) = 6 - 2x + 4 $$ This simplifies to: $$ 10 - 2x $$

Step 3 - Set up the simplified equation

Now we have the simplified equation: $$ -8x + 4 = 10 - 2x $$

Step 4 - Combine like terms

Add 2x to both sides to begin combining like terms: $$ -8x + 2x + 4 = 10 - 2x + 2x $$ This simplifies to: $$ -6x + 4 = 10 $$

Step 5 - Isolate the variable term

Subtract 4 from both sides to isolate the variable term: $$ -6x + 4 - 4 = 10 - 4 $$ This simplifies to: $$ -6x = 6 $$

Step 6 - Solve for x

Divide both sides by -6 to solve for x: $$ x = \frac{6}{-6} $$ This simplifies to: $$ x = -1 $$

Key Concepts

Distributive Property

Understanding the distributive property is crucial when solving linear equations. In this exercise, you distributed the constant 4 on the left side. This means you need to multiply 4 by each term inside the parentheses.

Let's see it step-by-step:

• If you have an expression such as \(4(-2x + 1)\), you multiply 4 by \(-2x\) and by 1.
• This gives us: \(4 \times -2x + 4 \times 1\).
• So, \(4 \times -2x\) becomes \(-8x\) and \(4 \times 1\) becomes 4.

Therefore, \(4(-2x + 1)\) results in \(-8x + 4\).

By applying this property correctly, you break down a more complex expression into simpler parts.

Combining Like Terms

Combining like terms helps streamline your equation. Like terms are terms that contain the same variable raised to the same power.

Consider the simplified equation we have:
\(-8x + 4 = 10 - 2x\).

• First, we need to get all the x terms on one side. Add \(2x\) to both sides.
• The equation becomes: \(-8x + 2x + 4 = 10 - 2x + 2x\).
• Simplify: \(-6x + 4 = 10\).

Our new equation, \(-6x + 4 = 10\), is simpler. We combined \(-8x\) and \(2x\) to get \(-6x\).

This process makes the equation easier to solve, setting up nicely for isolating variables.

Isolating Variables

Isolating the variable is a key step to solving for it. The aim is to get the variable term on one side of the equation by itself.

We have the equation \(-6x + 4 = 10\).

• Subtract 4 from both sides to isolate the variable term: \(-6x + 4 - 4 = 10 - 4\).
• This simplifies to: \(-6x = 6\).

This sets us up to solve for x easily.

By isolating the variable, you've made the equation straightforward, only needing one more step to find the value of x.

Simplifying Expressions

Simplifying expressions involves making an equation easier to read and solve.

We have \(-6x = 6\).

• To solve for x, divide both sides by -6: \(x = \frac{6}{-6}\).
• This simplifies to \(x = -1\).

Simplification clears unnecessary clutter and solves for x efficiently.

Always ensure expressions are as simple as possible for accurate, quick solutions.