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Chapter 2: Problem 45

Solve each equation with fraction coefficients. $$ \frac{5}{6} y-\frac{2}{3}=-\frac{3}{2} $$

Short Answer

Expert verified
y = -1

Step by step solution

01

Eliminate the fractions

To eliminate fractions, find the least common multiple (LCM) of the denominators. The denominators are 6, 3, and 2. The LCM of these numbers is 6. Multiply every term in the equation by 6.
02

Simplify the equation

After multiplying each term by 6, the equation becomes: \[ \frac{5}{6} y \times 6 - \frac{2}{3} \times 6 = -\frac{3}{2} \times 6 \] Simplify each term: \[ 5y - 4 = -9 \]
03

Isolate the variable term

Add 4 to both sides of the equation to move the constant term to the right side: \[ 5y - 4 + 4 = -9 + 4 \] This simplifies to: \[ 5y = -5 \]
04

Solve for y

Divide both sides of the equation by 5: \[ \frac{5y}{5} = \frac{-5}{5} \] This simplifies to: \[ y = -1 \]

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Key Concepts

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

Eliminating Fractions
When solving equations with fractions, starting by eliminating the fractions makes the equation much simpler. This can be done by finding a common multiple of all denominators involved. For example, in the equation \(\frac{5}{6} y-\frac{2}{3}=-\frac{3}{2}\), the denominators are 6, 3, and 2. The least common multiple (LCM) of these numbers is 6. By multiplying each term in the equation by this LCM, you can convert all fractional terms into whole numbers. This step greatly simplifies the problem and makes it easier to solve.
Least Common Multiple
The least common multiple (LCM) of a set of numbers is the smallest number that all the given numbers divide into without leaving a remainder. For example, for the denominators 6, 3, and 2, the LCM is 6. By multiplying the entire equation by this number, we can eliminate the fractions. Using the LCM ensures that no denominator remains in fractional form. Here鈥檚 how you do it in our example: Multiply each term in \(\frac{5}{6} y-\frac{2}{3}=-\frac{3}{2}\) by 6 to get \[5y - 4 = -9.\] This transforms the original equation into one that contains only whole numbers, which is much easier to handle.
Isolating the Variable
After eliminating fractions and simplifying, the next step is to isolate the variable. In our simplified equation \[5y - 4 = -9,\] the goal is to solve for y. First, move any constants to the other side of the equation by adding or subtracting. We add 4 to both sides to move the constant term to the right: \[5y - 4 + 4 = -9 + 4\] Simplifying this, we get \[5y = -5.\] Now, divide each term by the coefficient of y (which is 5 in this case) to isolate the variable: \[ \frac{5y}{5} = \frac{-5}{5}\] This simplifies to \[ y = -1.\] By isolating y, we have successfully solved for the variable.

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