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Magazine Chapter 2: Problem 96
Solve each equation. $$ -\frac{5}{6} x=30 $$
Short Answer
Expert verified
x = -36
Step by step solution
01
Isolate the variable
To solve the equation \(-\frac{5}{6} x = 30\), first isolate the variable x. This can be done by multiplying both sides of the equation by the reciprocal of \(-\frac{5}{6}\), which is \(-\frac{6}{5}\).
02
Multiply both sides
Multiply both sides of the equation by \(-\frac{6}{5}\): \(-\frac{5}{6} x \times -\frac{6}{5} = 30 \times -\frac{6}{5}\)
03
Simplify the left side
On the left side, the \(-\frac{5}{6}\) and \(-\frac{6}{5}\) terms will cancel each other out, leaving just x: \(x = 30 \times -\frac{6}{5}\)
04
Calculate the right side
Perform the multiplication on the right side: \(30 \times -\frac{6}{5} = -36\)
05
Solution
Thus, the solution to the equation is \(x = -36\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Isolating Variables
When solving linear equations like \(-\frac{5}{6} x = 30\), the first step often involves isolating the variable you are solving for, which is x in this case.
- To isolate x, you need to get rid of any coefficients or constants that are multiplied or added to x on one side of the equation.
- In the given equation, x is multiplied by \(-\frac{5}{6}\).
- By isolating the variable, you'll be left with x on one side of the equation, making it easier to solve.
Reciprocal Multiplication
Reciprocal multiplication is a key technique in solving linear equations. The reciprocal of a number is essentially 1 divided by that number.
\[ -\frac{5}{6} x \times -\frac{6}{5} = 30 \times -\frac{6}{5} \] This multiplication will cancel out \(-\frac{5}{6}\) on the left side, leaving us with x on its own.
- For a fraction, its reciprocal simply flips the numerator and the denominator. So, the reciprocal of \(-\frac{5}{6}\) is \(-\frac{6}{5}\).
- When you multiply a number with its reciprocal, the result is always 1. This is because the multiplication of a number by its reciprocal cancels out the fraction.
\[ -\frac{5}{6} x \times -\frac{6}{5} = 30 \times -\frac{6}{5} \] This multiplication will cancel out \(-\frac{5}{6}\) on the left side, leaving us with x on its own.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is crucial for solving equations efficiently. Cleaning up these expressions makes it easier to see the solution.
\[ -\frac{5}{6} x \times -\frac{6}{5} = x \] Leaving us with:
\[ x = 30 \times -\frac{6}{5} \] With x isolated, we now focus on multiplying the right side and simplifying it.
- In our example, once we perform the multiplication on both sides, we simplify the left side.
- The terms \(-\frac{5}{6}\) and \(-\frac{6}{5}\) cancel each other out because their product is 1.
\[ -\frac{5}{6} x \times -\frac{6}{5} = x \] Leaving us with:
\[ x = 30 \times -\frac{6}{5} \] With x isolated, we now focus on multiplying the right side and simplifying it.
Negative Reciprocals
Understanding negative reciprocals is important as it often appears in algebraic manipulations.
Finally, after solving, we end up with the solution:
\[ x = -36 \] Understanding these concepts helps solve linear equations efficiently and correctly.
- The negative reciprocal of a fraction is achieved by flipping the fraction and changing the sign. So, the negative reciprocal of \(\frac{5}{6}\) is \(-\frac{6}{5}\).
- In the equation, we used the negative reciprocal \(-\frac{6}{5}\) to make sure the negative sign is also effectively handled.
Finally, after solving, we end up with the solution:
\[ x = -36 \] Understanding these concepts helps solve linear equations efficiently and correctly.
