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Chapter 14: Problem 3

Solve. $$\frac{x+2}{4}-\frac{x-1}{5}=15$$

Short Answer

Expert verified
The short version of the solution is: To solve the equation \(\frac{x+2}{4}-\frac{x-1}{5}=15\), find the common denominator (20), eliminate the fractions, distribute, combine like terms, and solve for x, which gives \(x=286\).

Step by step solution

01

Identify the common denominator

The two denominators in the equation are 4 and 5. The least common multiple (LCM) of 4 and 5 is 20. So, let's set 20 as our common denominator.
02

Multiply each term by the common denominator

To eliminate the fractions in our equation, we will multiply each term by 20, which is our common denominator. \(20(\frac{x+2}{4}-\frac{x-1}{5})=20(15)\)
03

Distribute the 20 and simplify

When we distribute the 20, we get: \(\frac{20(x+2)}{4}-\frac{20(x-1)}{5}=20(15)\) Now, simplify the fractions: \((5)(x+2)-(4)(x-1)=300\)
04

Distribute and combine like terms

Apply the distributive property for both terms: \(5x+10-4x+4=300\) Now, combine the like terms: \(x+14=300\)
05

Solve for x

To solve for x, subtract 14 from both sides of the equation: \(x+14-14=300-14\) This simplifies to: \(x=286\) Therefore, the solution for the given equation is \(x=286\).

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Key Concepts

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

Common Denominator
When solving equations with fractions, finding a common denominator is an essential step. It allows us to eliminate the fractions and work with whole numbers, making the equation easier to solve. In this exercise, the denominators are 4 and 5. To find the least common denominator, find the smallest number that both 4 and 5 can divide into without leaving a remainder.
  • The multiples of 4 are: 4, 8, 12, 16, 20, ...
  • The multiples of 5 are: 5, 10, 15, 20, ...
The smallest multiple common to both lists is 20.
By establishing 20 as our common denominator, we can multiply each term in the equation by 20 to clear the fractions, simplifying our work moving forward.
Distributive Property
The distributive property is a useful tool in algebra that helps simplify expressions. It states that you can distribute a multiplier over terms inside parentheses. In our problem, after multiplying each term by the common denominator, 20, we have:\[20\left(\frac{x+2}{4}\right) - 20\left(\frac{x-1}{5}\right) = 20(15)\]Distributing allows us to simplify:\[\frac{20(x+2)}{4} - \frac{20(x-1)}{5} = 300\]Using the distributive property further simplifies to:\[5(x+2) - 4(x-1) = 300\]This step is critical because it transforms a potentially complex equation into a simpler form that is much easier to handle.
Combining Like Terms
After using the distributive property, the next step is to combine like terms. Like terms have the same variable raised to the same power, making them eligible to be added or subtracted together. From our expression, we gather:\[5x + 10 - 4x + 4 = 300\]To combine the like terms:
  • Group the terms with \(x\): \(5x - 4x\)
  • Group the constant terms: \(10 + 4\)
Simplify the expression by adding or subtracting the grouped terms:\[x + 14 = 300\]Combining like terms further reduces the equation, bringing us closer to finding the solution.
Fractions in Algebra
Working with fractions in algebra might seem intimidating, but with a few strategies, it becomes much more straightforward. Fractions often require finding a common denominator for addition or subtraction, as you've seen. Once denominators match, operations become manageable.
Multiplying by the common denominator eliminates fractions quickly, turning fractional equations into simpler linear equations. This method preserves the equation's balance while removing denominators. Once simplified, solving the equation is often as simple as basic arithmetic. Remember, mastering fractions in algebra opens the door to tackling more intricate math problems with confidence.

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