Problem 40 - Algebra A Combined Function

Question

Solve each equation. $$ 5(3-4 x)=9 $$

Step-by-Step Solution

Answer

The solution is \(x = 0.3\).

Step 1: Distribute the 5

Start by distributing the 5 into the expression inside the parentheses: \(5(3 - 4x)\). This means you multiply 5 by each term inside the parentheses: \(5 \times 3 - 5 \times 4x\). This simplifies to \(15 - 20x\).

Step 2: Write the equation

After the distribution, the equation \(5(3 - 4x) = 9\) becomes \(15 - 20x = 9\).

Step 3: Isolate the variable term

To isolate the term with \(x\), subtract 15 from both sides of the equation: \(15 - 15 - 20x = 9 - 15\), which simplifies to \(-20x = -6\).

Step 4: Solve for x

Divide both sides of the equation by -20 to solve for \(x\): \(-20x / -20 = -6 / -20\). This gives \(x = 3/10\).

Step 5: Simplify the solution (if necessary)

The fraction \(3/10\) is already in its simplest form, so the solution is \(x = 0.3\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It allows us to remove parentheses by distributing a multiplication over addition or subtraction within the parentheses. In the expression \(5(3 - 4x)\), the 5 is multiplied by each term inside the parentheses. So, you calculate \(5 \times 3\) and \(5 \times -4x\).

• This gives us \(15 - 20x\).
• Remember, if you distribute a negative number, it changes the signs of the terms inside the parentheses.
• This step simplifies complex equations, making them easier to manage and solve.

Linear Equations

Linear equations are equations that make a straight line when graphed. They are in the form of \(ax + b = c\), where \(x\) is the variable, and \(a\), \(b\), and \(c\) are constants. Linear equations often appear in everyday situations, like calculating expenses or distances.

• The goal with linear equations is to find the value of the variable that makes the equation true.
• Understanding the structure helps simplify the solving process, like moving terms or dividing numbers.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In our exercise, after distributing, we have a new equation: \(15 - 20x = 9\). Here, we need to isolate our variable, which is \(x\).

• Start by simplifying both sides as much as possible using arithmetic operations.
• Check that each step leads closer to isolating \(x\).

In this example, eliminating the constant term from the left by subtracting 15 from both sides allows us to focus on the variable part.

Isolation of Variable

The isolation of the variable is a critical step in solving equations. We manipulate the equation to get \(x\) alone on one side. In the example equation, \(-20x = -6\), dividing both sides by \(-20\) isolates \(x\).

• Performing the same operation on both sides keeps the equation balanced and unchanged in terms of truth.
• It transforms the problem from a complex one to a simple arithmetic calculation.
• In our case, it results in \(x = \frac{3}{10}\) or \(x = 0.3\).

Mastering this step is essential for solving more complex equations with ease.