Problem 27 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$9 x-6=-3(x+2)-24$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -2 \).

Step 1: Expand the equation

Begin by distributing the \(-3\) on the right side of the equation: \[-3(x + 2) = -3x - 6\]Insert it back into the equation, and it becomes: \[9x - 6 = -3x - 6 - 24\]

Step 2: Simplify the equation

Now, simplify the right side of the equation by combining like terms:\[-3x - 6 - 24 = -3x - 30\] The equation simplifies to: \[9x - 6 = -3x - 30\]

Step 3: Collect all x-terms on one side

Add \(+3x\) to both sides of the equation to get all the \(x\) terms on one side:\[9x + 3x - 6 = -3x + 3x - 30\]\[12x - 6 = -30\]

Step 4: Isolate the x-term

Add \(+6\) to both sides to isolate the \(x\) term:\[12x - 6 + 6 = -30 + 6\]\[12x = -24\]

Step 5: Solve for x

Divide both sides by \(+12\) to solve for \(x\):\[x = \frac{-24}{12}\]\[x = -2\]

Key Concepts

Distributive Property

The distributive property is a useful tool in algebra when you need to expand an expression. It allows you to multiply a single term by each term within a set of parentheses. For example, when faced with an expression like

• \(-3(x + 2)\)

you distribute the

• \(-3\)

through the terms inside the parentheses. This gives

• \(-3 \cdot x\) and \(-3 \cdot 2\), resulting in \(-3x - 6\).

Applying this concept helps in simplifying expressions and lays the groundwork for solving equations. Remember, when using the distributive property:

• Make sure each term inside the parentheses is multiplied by the term outside.
• Keep track of negative signs, as they can change the value of the expression.

Understanding the distributive property can simplify expressions and makes solving equations easier.

Combining Like Terms

Combining like terms is essential in simplifying algebraic expressions. When you have terms that look the same, i.e., they share the same variable raised to the same power, you can combine them by adding or subtracting their coefficients. For example:

• Consider the simplified equation \(9x - 6 = -3x - 30\).

On the right side,

• we identify \(-3x\) as a like term with the \(9x\) on the left side.

To simplify, rearrange so that all terms involving \(x\) are collected together:

• \(9x + 3x - 6\) simplifies to \(12x - 6\).

Important points to remember:

• Only terms with the exact same variable parts can be combined.
• Constant terms (numbers without variables) can be combined to simplify the expression further.

Combining like terms makes the expression easier to work with.

Solving for x

Solving for \(x\) means finding the value of the variable that makes the equation true. After simplifying and rearranging an equation, you'll often need to isolate \(x\) to find its value, as seen in our example:

• After isolating terms related to \(x\), we reached the equation \(12x = -24\).

This shows that to find the solution, you have to perform the inverse operation of multiplication:

• Divide both sides of the equation by \(12\).

This results in:

• \(x = \frac{-24}{12}\).

Key points about solving for \(x\):

• Always perform the opposite operation to isolate \(x\). For multiplication, divide; for addition, subtract.
• Simplify \(\frac{-24}{12}\) to get \(x = -2\).

Solving for \(x\) helps in determining the value that satisfies the equation.

Isolating Variables

Isolating variables is an important step in solving equations. It means getting the variable, like \(x\), alone on one side of the equation so we can solve for it directly. Here's how it works in our given equation:

• We arrived at \(12x = -24\) after simplification and combining like terms.

In order to isolate \(x\), we ensured all operations on the same side of \(x\) have been undone:

• Add \(+6\) to both sides to counteract the \(-6\) giving \(12x - 6 + 6 = -24 + 6\).
• At this point, we move to solve \(12x = -24\).

Tips for isolating variables:

• Consider reversing operations in the order opposite to PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).
• Verify each step to ensure no mistakes while balancing the equation.

This process is crucial for finding solutions to equations efficiently.