Problem 16 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$5(x+3)+4 x-3=-(2 x-4)+2$$

Step-by-Step Solution

Answer

x = -\frac{6}{11}

Step 1 - Distribute and Simplify

Distribute the 5 on the left side:\[5(x + 3) + 4x - 3 = 5x + 15 + 4x - 3\]Combine like terms:\[9x + 12\]On the right side, distribute the negative sign:\[-(2x - 4) + 2 = -2x + 4 + 2\]Combine like terms:\[-2x + 6\]Now the equation is:\[9x + 12 = -2x + 6\]

Step 2 - Move Variables to One Side

Add \(2x\) to both sides to move all \(x\) terms to one side:\[9x + 12 + 2x = -2x + 6 + 2x\]Simplify:\[11x + 12 = 6\]

Step 3 - Solve for x

Subtract 12 from both sides to move constants to the right side:\[11x + 12 - 12 = 6 - 12\]Simplify:\[11x = -6\]Divide both sides by 11 to solve for \(x\):\[x = \frac{-6}{11}\]

Key Concepts

Distributive Property

The distributive property is a key principle in algebra. It allows you to multiply a number by a sum or difference inside a set of parentheses. For instance, in the equation \(5(x + 3)\), you would distribute 5 to both \(x\) and 3. So, this becomes:
\(5(x + 3) = 5x + 15\).
This is helpful because it simplifies the equation, making it easier to solve. Always perform the distribution for every term inside the parentheses.

Combining Like Terms

When solving linear equations, combining like terms is essential. Like terms are terms that have the same variable raised to the same power. For example:
Combining terms in the expression \(5x + 15 + 4x - 3\) yields:
\(5x + 4x\) (both have the variable \(x\)) and \(15 - 3\) (both are constants).
Simplifying, this becomes:
\(9x + 12\).
Combining like terms makes equations simpler and more manageable.

Isolating Variables

Isolating variables means getting the variable you are solving for on one side of the equation. This is done so that you can solve the equation for that variable. In our example, after combining like terms, we have:
\(9x + 12 = -2x + 6\).
To isolate \(x\), add \(2x\) to both sides:
\(9x + 12 + 2x = -2x + 6 + 2x\), which simplifies to \(11x + 12 = 6\).
Then, move the constant term by subtracting 12 from both sides:
\(11x + 12 - 12 = 6 - 12\), which simplifies to \(11x = -6\).
Finally, divide by 11 to isolate \(x\):
\(x = \frac{-6}{11}\).

Basic Algebra

Basic algebra involves understanding and applying foundational concepts like variables, constants, and operations (addition, subtraction, multiplication, and division). When solving equations, follow these steps:

• Distribute constants.
• Combine like terms.
• Isolate the variable.
• Simplify your answer.

Practicing these steps will help you solve equations more effectively and build a strong mathematical foundation.