Problem 97 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{-3 x}{-2}+1=4 $$

Step-by-Step Solution

Answer

The solution is \( x = 2 \).

Step 1: Simplify the Fraction

Start by simplifying \( \frac{-3x}{-2} \). Notice that we can simplify this fraction: \( \frac{-3x}{-2} = \frac{3x}{2} \). This happens because dividing by a negative flips the sign.

Step 2: Isolate the Fraction

We start with the equation: \( \frac{3x}{2} + 1 = 4 \). To isolate the \( \frac{3x}{2} \) term, subtract 1 from both sides of the equation to get \( \frac{3x}{2} = 3 \).

Step 3: Eliminate the Fraction

To eliminate the fraction, multiply both sides of the equation by 2. This gives: \( 3x = 6 \).

Step 4: Solve for x

To solve for \( x \), divide both sides of the equation by 3. This results in \( x = 2 \).

Key Concepts

Isolating the variable

When you're solving an equation, isolating the variable means getting the variable all by itself on one side of the equation. This helps us find out what the variable is equal to. Think of it like solving a mystery and the variable is the final clue. You want it alone so you can see it clearly.
In our equation, after simplifying the fraction, we had: \[ \frac{3x}{2} + 1 = 4 \] The goal here is to get \( \frac{3x}{2} \) by itself. To do this, you need to subtract 1 from both sides to remove the +1 that's sitting there. It's just like balancing a seesaw—whatever you do to one side, you do to the other, to keep everything balanced. So after subtracting 1, we end up with: \[ \frac{3x}{2} = 3 \] Now, the variable is almost isolated. It's contained within a fraction, but we'll address that next. Remember, taking these clear, logical steps ensures we aren't leaving any clues out of sight!

Simplifying Fractions

Simplifying fractions is a crucial step when solving equations involving fractions. It helps make the equation easier to work with by reducing fractions to their simplest form. This can involve canceling factors or simplifying signs.
In the original problem, the fraction \( \frac{-3x}{-2} \) is present. When we look at a fraction like \( \frac{-3x}{-2} \), dividing by a negative turns the whole fraction to positive, because a negative divided by a negative equals a positive.
Thus, \( \frac{-3x}{-2} \) becomes \( \frac{3x}{2} \). Simplifying might seem small, but it's vital—it helps maintain an equation's balance and clarity.

• Negative signs can be tricky, remember that two negatives make a positive.
• Always simplify before moving ahead, it lays down a clearer path for later steps.

Simplifying can greatly impact the ease of finding the final solution, so it's a habit to nurture!

Solving Linear Equations

Solving linear equations is all about finding the value of the variable that makes the equation true. When dealing with linear equations involving fractions, our first task is to clear the equation from fractions to make it more straightforward to solve.
Here's what happened in our example:

• After simplifying the fraction, we isolated \( \frac{3x}{2} \) by subtracting 1 to get \( \frac{3x}{2} = 3 \).
• Next, we eliminated the fraction by multiplying the whole equation by 2, to clear out the divisor: \( 3x = 6 \).

This process leaves us with a simple linear equation without fractions. Now, to find \( x \), we divide both sides by 3. Simple division gives us the solution \( x = 2 \).
Always remember: linear equations involve straightforward operations. Breaking the equation down into small steps makes solving them less intimidating and easier to manage. Keep these steps in mind as you practice, and with time, they'll become second nature!