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Chapter 2: Problem 68

Solve each equation. Check all solutions. $$\frac{5 x}{6}+\frac{3}{5}=3$$

Short Answer

Expert verified
The solution of the equation is \( x = \frac{72}{25} \)

Step by step solution

01

Simplify the Equation

The first stage involves rewriting the equation \(\frac{5x}{6}+\frac{3}{5}=3\) in order to get rid of fractions. This is done by finding the least common multiple of the denominators to eliminate the fractions. The least common multiple between 6 and 5 is 30. So, multiply each term by 30. This gives: \n\[30(\frac{5x}{6}) + 30(\frac{3}{5}) = 30(3)\] Simplifying it leads to: \n\[25x + 18 = 90\]
02

Isolate the Variable

To isolate the variable x, subtract 18 from each side of the equation: \n\[25x = 90 - 18 \] This simplifies to: \n\[25x = 72\] Now divide both sides by 25 to solve for x. \n\[x = \frac{72}{25}\]
03

Check the Solution

To verify the solution, substitute the calculated value of x back into the original equation: \n\[\frac{5(\frac{72}{25})}{6} + \frac{3}{5} = 3\] Simplifying the left hand side, it indeed equals to the right hand side, confirming that the solution is correct.

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Key Concepts

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

Simplifying Equations
When faced with algebraic equations, simplifying is essential to making the problem more manageable. Simplification involves several strategies like combining like terms, and in the case of fractions, finding a common denominator to eliminate them. For example, consider the equation \[\frac{5x}{6} + \frac{3}{5} = 3\].

To simplify this equation, you would look for ways to combine the fractions into a single term without fractions. This leads to a clearer pathway towards solving for the variable.
Least Common Multiple
In simplifying equations, the least common multiple (LCM) of denominators is used to eliminate fractions. The LCM is the smallest number that is a multiple of both denominators. In our equation \[\frac{5x}{6} + \frac{3}{5} = 3\],
the denominators are 6 and 5.

Their LCM is 30, and by multiplying every term by the LCM, you effectively clear the fractions, making the equation easier to work with. This strategy ensures that you're working with integers, which are simpler to handle than fractions.
Isolating the Variable
Step towards solving an algebraic equation involves isolating the variable. This means to get the variable on one side of the equation and numbers on the other. Using the simplified form of our exercise equation, \[25x + 18 = 90\],
we subtract 18 from both sides to isolate the term with the variable \[25x\].

From there, dividing both sides by the coefficient of the variable (in this case, 25) yields the value of x. Isolating the variable is a fundamental skill in algebra that allows you to solve equations effectively.
Checking Solutions
After finding a solution to an equation, it's essential to check that your answer is correct. This validation step involves substituting the value back into the original equation and seeing if the left-hand side equals the right-hand side. If they match, the solution is correct.

For instance, in our exercise where \[x = \frac{72}{25}\],
we substitute this value back into the initial equation to verify its correctness. This step prevents the overlook of potential errors and ensures the integrity of the solution.

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