Problem 32 - Algebra 1

Question

Solve the equation. $$-\frac{4}{9}(2 x-4)=48$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -52 \)

Step 1: Distribute the factor

Multiply \( -\frac{4}{9} \) through the parentheses to get \( -\frac{4}{9}*2x + \frac{4}{9}*4 = 48 \)

Step 2: Simplify each side

This simplifies to \( -\frac{8}{9}x + \frac{16}{9} = 48 \)

Step 3: Isolate the variable term

Subtract \( \frac{16}{9} \) from both sides of the equation to isolate the variable term on one side. This results in \( -\frac{8}{9}x = 48 - \frac{16}{9} \)

Step 4: Simplify the right side

After simplification, we get \( -\frac{8}{9}x = \frac{416}{9} \)

Step 5: Solve for x

Finally, to solve for x, multiple both sides by \( -\frac{9}{8} \) to get \( x = -\frac{9}{8} * \frac{416}{9} \)

Step 6: Simplify the answer

Simplify to get the solution \( x = -52 \)

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps you simplify expressions and equations. This property is crucial when dealing with expressions that involve parentheses. The distributive property states that when you multiply a single term by terms inside parentheses, you should multiply the outside term by each term inside the parentheses separately. This makes working with complex expressions much easier.

For instance, consider the initial step in our equation: \(-\frac{4}{9}(2x-4) = 48\). We apply the distributive property by multiplying \(-\frac{4}{9}\) with each term inside the parentheses \((2x\) and \(-4)\):

• \(-\frac{4}{9} \times 2x = -\frac{8}{9}x\)
• \(-\frac{4}{9} \times -4 = \frac{16}{9}\)

This simplifies our expression to \(-\frac{8}{9}x + \frac{16}{9}\). Using the distributive property ensures accuracy in the manipulation of algebraic expressions.

Isolating the Variable

Isolating the variable is a key step in solving equations. The goal is to get the variable by itself on one side of the equation. This allows us to determine its value by performing operations on both sides of the equation.

• Start by simplifying the equation as much as possible.
• Add, subtract, multiply, or divide both sides of the equation to move any constants or coefficients away from the variable.

In our example, we start with the expression \(-\frac{8}{9}x + \frac{16}{9} = 48\). To isolate \(x\), we first need to eliminate the \(\frac{16}{9}\). We do this by subtracting \(\frac{16}{9}\) from both sides:

• Left side: \(-\frac{8}{9}x\)
• Right side: \(48 - \frac{16}{9} = \frac{416}{9}\)

Now, the variable term \(-\frac{8}{9}x\) stands alone on one side, making it easier to solve for \(x\).

Simplifying Equations

Simplifying equations involves reducing an equation to its most basic form, making it easier to solve. This process typically involves combining like terms, clearing fractions, and ensuring each side of the equation is as simple as possible.

• Combine like terms when possible (terms with the same variables to the same power).
• Eliminate fractions by finding a common denominator or multiplying through by the reciprocal.

In our problem, simplifying started with applying the distributive property and combining terms to simplify both sides of the equation: \(-\frac{8}{9}x + \frac{16}{9} = \frac{416}{9}\). After isolating the variable term, further simplification was necessary to solve for \(x\). We multiplied both sides by the reciprocal of \(-\frac{8}{9}\) to find the solution:

• Multiply by \(-\frac{9}{8}\), resulting in \(x = -\frac{9}{8} \times \frac{416}{9}\)
• Simplify the multiplication to get \(x = -52\)

Simplifying ensures the solution is both accurate and easy to understand.