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Chapter 1: Problem 57

Solve each equation, and check the solution. \(\frac{6}{5} x=-1\)

Short Answer

Expert verified
x = -\frac{5}{6}. Verified by substitution.

Step by step solution

01

- Isolate the Variable

To isolate the variable x, multiply both sides of the equation by the reciprocal of \(\frac{6}{5}\). The reciprocal of \(\frac{6}{5}\) is \(\frac{5}{6}\).
02

- Multiply Both Sides

Multiply both sides of the equation by \(\frac{5}{6}\): \(x = -1 \times \frac{5}{6}\).
03

- Simplify

Simplify the right side: \(x = -\frac{5}{6}\).
04

- Check the Solution

Substitute \(-\frac{5}{6}\) back into the original equation to verify: \(\frac{6}{5} \times -\frac{5}{6} = -1\). Simplify the left side to get \(-1\), which matches the right side of the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

isolating the variable
Isolating the variable is a fundamental step in solving linear equations. It means getting the variable (like x) alone on one side of the equation. This helps you find its value.
To isolate the variable, you often need to undo operations around it. If something is added, you subtract it. If something is multiplied, you divide it. The goal is to get the variable by itself.

For instance, in the equation \(\frac{6}{5} x = -1\), you want to isolate the variable x. Multiply both sides by the reciprocal of \(\frac{6}{5}\) to make this happen. That way, x will stand alone on one side.
reciprocal
Reciprocal is a concept that frequently pops up in equations involving fractions. The reciprocal of a number is what you multiply that number by to get 1. Simply put, flip the numerator and denominator.
  • The reciprocal of 2/3 is 3/2.
  • The reciprocal of 5 is 1/5.
  • The reciprocal of 1/4 is 4.

In our equation \(\frac{6}{5} x = -1\), the reciprocal of \(\frac{6}{5}\) is \(\frac{5}{6}\). We multiply both sides by \(\frac{5}{6}\) to isolate x. This move transforms the equation into x on one side and helps solve it.
multiplication property of equality
This property says that you can multiply both sides of an equation by the same number without changing the equation's balance.
It ensures equality remains intact. This is vital when isolating variables.
  • If you have \(a = b\), multiplying both sides by the same number still ensures \(a \times c = b \times c\).
  • In the equation \(\frac{6}{5} x = -1\), you use this property to multiply both sides by \(\frac{5}{6}\) and keep the balance.

This step maintains the equation's integrity while simplifying it.
simplifying fractions
Simplifying fractions makes expressions easier to understand and work with. It involves reducing the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor.
  • The fraction \( \frac {10}{20} \) simplifies to \(\frac{1}{2}\).
  • In solving \(\frac{6}{5} x = -1\), the resulting fraction after multiplying by \(\frac{5}{6}\) is \(\frac{-5}{6}\), which is already simplified.

Always look for ways to simplify fractions in your final answers for clarity and precision.

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