Problem 23 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$-4(2 x-6)+8 x=5 x+24+x$$

Step-by-Step Solution

Answer

x = 0.

Step 1 - Distribute on the left side

Distribute the -4 across the terms inside the parentheses: -4(2x - 6) + 8x = 5x + 24 + x becomes -8x + 24 + 8x = 5x + 24 + x.

Step 2 - Simplify the equation by combining like terms

Combine the terms on both sides: -8x + 8x + 24 = 5x + x + 24 which simplifies to 24 = 6x + 24.

Step 3 - Isolate the variable

Move the constant term on the right side by subtracting 24 from both sides: 24 - 24 = 6x + 24 - 24 which simplifies to 0 = 6x.

Step 4 - Solve for x

Divide both sides by 6: 0 / 6 = 6x / 6 which simplifies to x = 0.

Key Concepts

Distribution in Algebra

When you see parentheses in an algebraic expression, you often need to distribute to simplify the expression. Distribution involves multiplying each term inside the parentheses by the term outside. For example, in the equation \[-4(2x - 6) + 8x = 5x + 24 + x\], you distribute the -4 to both 2x and -6:\[-4 \times 2x = -8x\]\[-4 \times -6 = 24\]Which turns the equation into: \[-8x + 24 + 8x = 5x + 24 + x\]Properly distributing terms is key to simplifying equations.

Combining Like Terms

After distribution, look for like terms to simplify further. Like terms are terms with the same variable raised to the same power. In the example \[-8x + 24 + 8x = 5x + 24 + x\],both \[-8x\] and \[8x\] are like terms, and \[5x\] and \[x\] are like terms. Combine them to simplify the equation:\[-8x + 8x + 24 = 6x + 24\]Here, \[-8x + 8x\] cancel each other out, simplifying to \[24 = 6x + 24\].

Isolating Variables

To isolate the variable, you need to get the variable term on one side of the equation and the constants on the other side. Starting from \[24 = 6x + 24\], you subtract 24 from both sides:\[24 - 24 = 6x + 24 - 24\],which simplifies to:\[0 = 6x\].Isolating the variable makes it easier to solve the equation.

Solving for x

Once the variable is isolated, solve for \[x\]. From \[0 = 6x\], divide both sides by \[6\]:\[0 / 6 = 6x / 6\], which simplifies to: \[x = 0\].Now, you have found the value of \[x\]. Remember, each phase brings you closer to the solution.