Problem 136 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{3 x}{7}=\frac{-3 x}{7}+12 $$

Step-by-Step Solution

Answer

The solution is \(x = 14\).

Step 1: Eliminate Fractions

To eliminate fractions, multiply both sides of the equation by 7. This simplifies the equation by removing the denominator.\[7 \times \frac{3x}{7} = 7 \times \left(\frac{-3x}{7} + 12 \right)\]Simplifying gives:\[3x = -3x + 84\]

Step 2: Combine Like Terms

Add \(3x\) to both sides to combine like terms and solve for \(x\):\[3x + 3x = 84\]This simplifies to:\[6x = 84\]

Step 3: Solve for x

Divide both sides of the equation by 6 to solve for \(x\):\[\frac{6x}{6} = \frac{84}{6}\]Simplifying gives:\[x = 14\]

Key Concepts

Eliminating Fractions

Fractions in equations can often make things seem more complicated than they are. To simplify and solve equations with fractions, the first step is to eliminate them.
A straightforward method to do this is to multiply every term on both sides of the equation by the denominator of the fraction. This process removes the fractions by cancelling them out.

• First, identify the denominator. In our example, it's 7.
• Multiply every part of the equation by this number (7 in our case).
• This action will cancel out the fractions and simplify the equation considerably.

In our example, after multiplying every term by 7, we have:\[7 \times \frac{3x}{7} = 7 \times \left(\frac{-3x}{7} + 12 \right)\]This simplifies to:\[3x = -3x + 84\] Once the fractions are gone, the equation looks much simpler and is easier to work with in the subsequent steps.

Combining Like Terms

Once fractions are eliminated, the next step is to tidy up the equation by combining like terms. Like terms are terms in an equation that have the same variable raised to the same power.
For example, in our equation \(3x = -3x + 84\), \(3x\) and \(-3x\) are like terms because they both have the variable \(x\).

• Move terms with the variable on one side of the equation. In our case, add \(3x\) to both sides.
• By doing this, you end up with all variable terms on one side and numbers on the other.
• This simplifies the equation and makes it easier to solve.

In our example, adding \(3x\) to both sides leads us to:\[3x + 3x = 84\]Simplifying gives:\[6x = 84\] At this stage, the equation is clean, with like terms combined, setting the stage for isolating the variable.

Isolation of Variable

Finally, to solve the equation, we need to isolate the variable. Isolating means getting \(x\) alone on one side of the equation.
This will give us the solution to the equation.
Follow these steps:

• Look at the equation after combining like terms, which is \(6x = 84\).
• To isolate \(x\), divide every term by the coefficient of \(x\). Here, the coefficient is 6.
• This division will straightforwardly give the value of \(x\).

Dividing both sides of the equation \(6x = 84\) by 6 results in:\[\frac{6x}{6} = \frac{84}{6}\]Simplifying gives:\[x = 14\] Now, you've successfully solved for \(x\) by isolating it. This same process can be applied to many equations where a variable needs to be solved.