Problem 94 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{-3 x}{7}-4=4 $$

Step-by-Step Solution

Answer

The solution is \( x = -\frac{56}{3} \).

Step 1: Isolate the Linear Term

To solve the equation \( \frac{-3x}{7} - 4 = 4 \), we need to isolate the linear term \( \frac{-3x}{7} \). We start by adding 4 to both sides of the equation to remove the constant term from the left side: \[ \frac{-3x}{7} - 4 + 4 = 4 + 4 \] This simplifies to: \[ \frac{-3x}{7} = 8 \]

Step 2: Eliminate the Fraction

Next, we eliminate the fraction by multiplying both sides of the equation by 7. This will clear the denominator: \[ 7 \times \frac{-3x}{7} = 8 \times 7 \] The equation simplifies to: \[ -3x = 56 \]

Step 3: Solve for x

Now, we solve for \( x \) by dividing both sides of the equation by -3 to isolate \( x \): \[ x = \frac{56}{-3} \] This gives us: \[ x = -\frac{56}{3} \] Thus, the solution to the equation is \( x = -\frac{56}{3} \).

Key Concepts

Isolate Linear Term

In order to solve a linear equation like \( \frac{-3x}{7} - 4 = 4 \), the first task is to isolate the linear term. The linear term here is \( \frac{-3x}{7} \). To begin the isolation process, you need to get rid of any constant values that are on the same side as the linear term.

For our exercise, we have the number \(-4\) alongside the linear term. The goal is to have the linear term by itself on one side of the equation. We can achieve this by adding \(4\) to both sides. This is known as applying the inverse operation because \(-4 + 4 = 0\).

This simplifies the equation to \( \frac{-3x}{7} = 8 \). At this stage, we have successfully isolated the linear term, which is a crucial step in solving the equation effectively.

Eliminate Fraction

Once the linear term is isolated, the next step is to eliminate the fraction. Fractions can often make equations seem more complex than they are, so it’s useful to get rid of them early in the process.

In our exercise, the linear term \( \frac{-3x}{7} \) has a denominator of 7. We eliminate this fraction by multiplying every term in the equation by 7, which is the denominator.

Here's what that process looks like:

• Multiply the fraction \( \frac{-3x}{7} \) by 7 to get \(-3x\).
• Multiply the right side of the equation, which is \(8\), by 7. This results in \(56\).

By multiplying both sides of the equation, the equation becomes \(-3x = 56\).

Removing the fraction simplifies the equation and makes it easier to work with, leading to a simpler final step.

Solve for Variable

Now that the equation is simplified to \(-3x = 56\), the last step is to solve for the variable \(x\). This means getting \(x\) by itself on one side of the equation.

Currently, \(x\) is being multiplied by \(-3\). To isolate \(x\), we perform the inverse operation by dividing both sides by \(-3\). This action counteracts the multiplication, working to leave \(x\) by itself.

The division looks like this:

• Divide \(-3x\) by \(-3\) to get \(x\).
• Divide 56 by \(-3\) to yield \(\frac{56}{-3}\).

After calculation, we have \(x = -\frac{56}{3}\).

This step ensures that we've isolated \(x\) and figured out its exact value, effectively solving the problem with clarity and precision.