Problem 41 - Algebra for College Students

Question

Solve each equation for the indicated variable. \(3(x-2 y)=4\) for \(x\)

Step-by-Step Solution

Answer

\(x = \frac{6y + 4}{3}\)

Step 1: Distribute the constant

The given equation is \(3(x-2y) = 4\). To eliminate the parentheses, distribute the constant \(3\) to each term inside the parentheses. This results in \(3x - 6y = 4\).

Step 2: Isolate the variable term

We need to solve for \(x\), so first isolate the \(x\) term on one side of the equation. Add \(6y\) to both sides to get \(3x = 6y + 4\).

Step 3: Solve for the variable

To solve for \(x\), divide each term in the equation by \(3\) to isolate \(x\). This gives \(x = \frac{6y + 4}{3}\).

Key Concepts

Understanding the Distributive Property

The distributive property is an important skill when solving equations, especially those involving parentheses.
It allows us to simplify expressions by distributing a constant across terms inside parentheses.
If you have an equation like \(a(b + c)\), the distributive property tells us that it equals \(ab + ac\).
In the exercise, we had \(3(x - 2y) = 4\). By applying the distributive property, we multiplied \(3\) by each term inside the parentheses:

• Multiply \(3\) by \(x\) to get \(3x\)
• Multiply \(3\) by \(-2y\) to get \(-6y\)

After distribution, the equation becomes \(3x - 6y = 4\).
Mastering the distributive property is crucial as it paves the way for isolating variables and solving complex equations.

Isolation of Variables Explained

The key to solving equations is often in isolating the variable you're solving for on one side of the equation.
Isolation of variables involves rearranging the equation to get the variable alone on one side.
In our example, we started with \(3x - 6y = 4\) after applying the distributive property.
To isolate \(x\), we needed to remove the term \(-6y\) from the left side.
This is done by adding \(6y\) to both sides, resulting in:

• Left side becomes \(3x\)
• Right side becomes \(6y + 4\)

Now, the equation reads \(3x = 6y + 4\). The variable \(x\) is closer to being isolated.
Breaking the process into small steps helps prevent errors and makes challenging problems manageable.

Breaking Down Linear Equations

Linear equations are equations of the first degree, which means they contain no variable terms with exponents higher than one.
They often look like \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
Our equation \(3(x-2y)=4\) falls into this category once simplified to \(3x - 6y = 4\).
The goal with linear equations is to find the value of the variable that makes the equation true.
After isolating the variable \(x\) (3x = 6y + 4), the final step is to solve for \(x\).
This is achieved by performing inverse operations to eliminate the coefficient \(3\) from \(3x\).
We do this by dividing every term by \(3\), leading to \(x = \frac{6y + 4}{3}\).
Solving linear equations involves a straightforward set of procedures that, once understood, are easy to apply to countless scenarios.

• The solution provides a relationship between \(x\) and \(y\), showcasing the connection within linear equations.
• By practicing, you'll gain confidence in handling more complex equations by mastering fundamentals like these.