Problem 132 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{-5 x}{7}=\frac{2 x}{7} $$

Step-by-Step Solution

Answer

The solution is x = 0.

Step 1: Identify Common Terms

Notice that both sides of the equation have the common denominator of 7. We can eliminate the denominators by multiplying each side of the equation by 7.

Step 2: Eliminate the Fractions

Multiply every term by 7 to get rid of the denominator:\[7 \times \left( \frac{-5x}{7} \right) = 7 \times \left( \frac{2x}{7} \right)\]This simplifies to:\[-5x = 2x\]

Step 3: Move All Terms to One Side

To isolate the variable on one side, subtract 2x from both sides:\[-5x - 2x = 0\]This simplifies to:\[-7x = 0\]

Step 4: Solve for the Variable

Divide both sides of the equation by -7 to solve for x:\[\frac{-7x}{-7} = \frac{0}{-7}\]This simplifies to:\[x = 0\]

Step 5: Verify the Solution

Substitute x = 0 back into the original equation to check:\[\frac{-5(0)}{7} = \frac{2(0)}{7}\]This results in 0 = 0, confirming the solution is correct.

Key Concepts

Eliminating Fractions

When dealing with algebraic equations, especially those involving fractions, the first goal is often to eliminate these fractions. Fractions can complicate the process of solving equations, so getting rid of the denominators at the earliest step simplifies the problem. In our example, both sides of the equation are fractions with a denominator of 7:

• Multiply both sides of the equation by the denominator 7.
• This cancels out the fractions, leaving behind only the numerators.

After multiplying by 7, we are left with a much simpler equation: \[-5x = 2x\].This technique of eliminating the fractions is a common strategy to tackle equations with fractional components.

Isolation of Variables

The core idea behind solving any equation is to isolate the variable, which means to get the variable alone on one side of the equation. In our new, simplified equation \(-5x = 2x\), we want to rearrange terms to isolate \(x\). Here's how to do it:

• Subtract \(2x\) from both sides to consolidate variable terms on one side.
• This results in \(-5x - 2x = 0\).

By moving terms strategically, we bring ourselves closer to finding the value of \(x\). It's a systematic process that often involves addition or subtraction to "move" terms around.

Solving Linear Equations

Linear equations involve variables raised only to the first power, as seen with \(-7x = 0\). Solving them often boils down to simple arithmetic once the variable is isolated.

• Here, divide both sides by \(-7\) to solve for \(x\).
• The result is \(x = 0\), which shows the variable's value.

This solution process can be applied to any linear equation. By focusing on isolating the variable and then solving it, you can find the solution efficiently.

Fractions in Equations

Fractions can be intimidating, but they don't have to be. When you encounter them in equations, remember:

• Identify a common denominator—this simplifies comparison and elimination.
• Use multiplication to eliminate fractions early on.
• Treat the equation just like any other by performing operations to isolate variables.

By understanding the basic operations needed to handle fractions, and knowing how to switch them into whole-number terms early, you make solving equations much more manageable. Always verify the solution by substituting it back into the original equation, which ensures the manipulation was correct.