Problem 24 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$-8(3 x+4)+6 x=4(x-8)+4 x$$

Step-by-Step Solution

Answer

x = 0

Step 1: Distribute the constants

First, distribute the term -8 to both 3x and 4 in -8(3x + 4). Similarly, distribute the 4 on the right side of the equation to (x-8). The equation then becomes -24x - 32 + 6x = 4x - 32 + 4x.

Step 2: Combine like terms

Combine like terms on both sides of the equation. On the left side, combine -24x with 6x to get -18x -32. On the right side, combine 4x and 4x to get 8x - 32. The equation now looks like this: -18x - 32 = 8x - 32.

Step 3: Move constants to one side

Add 32 to both sides to move the constant term to one side. This results in the equation: -18x = 8x.

Step 4: Isolate the variable

Subtract 8x from both sides to isolate the variable x . The equation becomes -26x = 0.

Step 5: Solve for x

Finally, divide by -26 to find the value of x . The solution is x = 0.

Key Concepts

Distributive Property

The distributive property is a useful tool in algebra. This property allows us to distribute a factor across terms inside parentheses. For instance, if we have something like \(a(b + c)\), we can rewrite it as \(ab + ac\).

In the given equation, -8 is distributed to both 3x and 4, giving us -24x and -32. Likewise, the 4 on the right side of the equation is distributed to both x and -8, resulting in 4x and -32. This helps to simplify the equation:

\[ -8(3x+4) + 6x = 4(x-8) + 4x \]

Becomes:

\[ -24x - 32 + 6x = 4x - 32 + 4x \]

Combining Like Terms

Combining like terms means merging terms that have the same variable raised to the same power. This helps to simplify the equation further and makes it easier to solve.

In our simplified equation from the first step, we have -24x and 6x on the left side, and 4x and 4x on the right side:

\[ -24x - 32 + 6x = 4x - 32 + 4x \]

We combine these like terms on each side:

-24x + 6x simplifies to -18x, and 4x + 4x simplifies to 8x. Now our equation looks like this:

\[ -18x - 32 = 8x - 32 \]

Isolating Variables

Isolating the variable is all about getting the variable on one side of the equation by itself. This is a crucial step in solving equations, as it allows us to determine the value of the variable.

In our equation, we want to isolate x. The initial equation after combining like terms looks like this:

\[ -18x - 32 = 8x - 32 \]

First, we add 32 to both sides to move the constants to one side:

\[ -18x - 32 + 32 = 8x - 32 + 32 \]

Simplifying, this gives us:

\[ -18x = 8x \]

Next, we subtract 8x from both sides to isolate the x variable:

\[ -18x - 8x = 8x - 8x \]

This simplifies to:

\[ -26x = 0 \]

Basic Algebra

Basic algebra involves foundational operations such as addition, subtraction, multiplication, and division to solve equations. These concepts lay the groundwork for more advanced mathematical topics.

In the final step of our solution with the equation \(-26x = 0\), we divide both sides by -26 to solve for x:

\[ \frac{-26x}{-26} = \frac{0}{-26} \]

Simplifying this, we get:

\[ x = 0 \]

As shown, the value of x that satisfies the original equation is 0. This workflow demonstrates how combining the principles of distributive property, combining like terms, isolating variables, and basic algebra can effectively solve linear equations.