Solve The Equation:$\frac{x+1}{2}+\frac{x+6}{3}=5$

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Introduction

In this article, we will delve into the world of algebra and solve a linear equation involving fractions. The equation we will be solving is x+12+x+63=5\frac{x+1}{2}+\frac{x+6}{3}=5. This equation may seem daunting at first, but with a clear understanding of the steps involved, we can break it down and find the solution.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at what it represents. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, xx) is 1. The equation involves two fractions, x+12\frac{x+1}{2} and x+63\frac{x+6}{3}, which are added together and set equal to 5.

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is 6. This will allow us to clear the fractions and work with whole numbers.

\frac{x+1}{2}+\frac{x+6}{3}=5

Multiplying both sides by 6 gives us:

6\left(\frac{x+1}{2}+\frac{x+6}{3}\right)=6(5)

Step 2: Distribute the 6 to Each Fraction

Now that we have multiplied both sides by 6, we can distribute the 6 to each fraction. This will allow us to simplify the equation and get rid of the fractions.

6\left(\frac{x+1}{2}\right)+6\left(\frac{x+6}{3}\right)=30

Step 3: Simplify the Fractions

Now that we have distributed the 6 to each fraction, we can simplify the fractions. To simplify a fraction, we divide the numerator and denominator by their greatest common divisor (GCD).

3(x+1)+2(x+6)=30

Step 4: Expand and Combine Like Terms

Now that we have simplified the fractions, we can expand and combine like terms. This will allow us to simplify the equation further and get closer to the solution.

3x+3+2x+12=30

Step 5: Combine Like Terms

Now that we have expanded and combined like terms, we can combine like terms further. This will allow us to simplify the equation even further and get closer to the solution.

5x+15=30

Step 6: Subtract 15 from Both Sides

Now that we have combined like terms, we can subtract 15 from both sides of the equation. This will allow us to isolate the variable xx and find the solution.

5x=15

Step 7: Divide Both Sides by 5

Finally, we can divide both sides of the equation by 5 to find the solution. This will give us the value of xx that satisfies the equation.

x=3

Conclusion

In this article, we have solved the equation x+12+x+63=5\frac{x+1}{2}+\frac{x+6}{3}=5 using a step-by-step approach. We have multiplied both sides by the least common multiple (LCM) of the denominators, distributed the 6 to each fraction, simplified the fractions, expanded and combined like terms, combined like terms further, subtracted 15 from both sides, and finally divided both sides by 5 to find the solution. The solution to the equation is x=3x=3.

Tips and Tricks

When solving equations involving fractions, it's essential to remember the following tips and tricks:

  • Multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.
  • Distribute the 6 to each fraction to simplify the equation.
  • Simplify the fractions by dividing the numerator and denominator by their greatest common divisor (GCD).
  • Expand and combine like terms to simplify the equation further.
  • Combine like terms further to simplify the equation even more.
  • Subtract 15 from both sides to isolate the variable xx.
  • Divide both sides by 5 to find the solution.

By following these tips and tricks, you can solve equations involving fractions with ease and confidence.

Frequently Asked Questions

Here are some frequently asked questions about solving equations involving fractions:

  • Q: What is the least common multiple (LCM) of the denominators? A: The least common multiple (LCM) of the denominators is the smallest number that both denominators can divide into evenly.
  • Q: How do I distribute the 6 to each fraction? A: To distribute the 6 to each fraction, multiply the numerator and denominator of each fraction by 6.
  • Q: How do I simplify the fractions? A: To simplify the fractions, divide the numerator and denominator by their greatest common divisor (GCD).
  • Q: How do I expand and combine like terms? A: To expand and combine like terms, multiply the terms together and combine like terms.
  • Q: How do I combine like terms further? A: To combine like terms further, multiply the terms together and combine like terms.
  • Q: How do I subtract 15 from both sides? A: To subtract 15 from both sides, subtract 15 from both sides of the equation.
  • Q: How do I divide both sides by 5? A: To divide both sides by 5, divide both sides of the equation by 5.

By following these tips and tricks, you can solve equations involving fractions with ease and confidence.

Final Thoughts

Solving equations involving fractions can seem daunting at first, but with a clear understanding of the steps involved, it can be a straightforward process. By following the tips and tricks outlined in this article, you can solve equations involving fractions with ease and confidence. Remember to multiply both sides by the least common multiple (LCM) of the denominators, distribute the 6 to each fraction, simplify the fractions, expand and combine like terms, combine like terms further, subtract 15 from both sides, and finally divide both sides by 5 to find the solution. With practice and patience, you can become a master of solving equations involving fractions.

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Introduction

Solving equations involving fractions can be a challenging task, but with the right guidance, it can be a straightforward process. In this article, we will answer some of the most frequently asked questions about solving equations involving fractions.

Q: What is the least common multiple (LCM) of the denominators?

A: The least common multiple (LCM) of the denominators is the smallest number that both denominators can divide into evenly. For example, if the denominators are 2 and 3, the LCM is 6.

Q: How do I distribute the 6 to each fraction?

A: To distribute the 6 to each fraction, multiply the numerator and denominator of each fraction by 6. For example, if the fraction is x+12\frac{x+1}{2}, you would multiply the numerator and denominator by 6 to get 6(x+1)12\frac{6(x+1)}{12}.

Q: How do I simplify the fractions?

A: To simplify the fractions, divide the numerator and denominator by their greatest common divisor (GCD). For example, if the fraction is 6(x+1)12\frac{6(x+1)}{12}, you would divide the numerator and denominator by 6 to get x+12\frac{x+1}{2}.

Q: How do I expand and combine like terms?

A: To expand and combine like terms, multiply the terms together and combine like terms. For example, if the equation is 3x+3+2x+12=303x+3+2x+12=30, you would multiply the terms together to get 5x+15=305x+15=30 and then combine like terms to get 5x=155x=15.

Q: How do I combine like terms further?

A: To combine like terms further, multiply the terms together and combine like terms. For example, if the equation is 5x=155x=15, you would multiply the terms together to get 5x=155x=15 and then combine like terms to get x=3x=3.

Q: How do I subtract 15 from both sides?

A: To subtract 15 from both sides, subtract 15 from both sides of the equation. For example, if the equation is 5x+15=305x+15=30, you would subtract 15 from both sides to get 5x=155x=15.

Q: How do I divide both sides by 5?

A: To divide both sides by 5, divide both sides of the equation by 5. For example, if the equation is 5x=155x=15, you would divide both sides by 5 to get x=3x=3.

Q: What if I have a fraction with a variable in the denominator?

A: If you have a fraction with a variable in the denominator, you will need to multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fraction. For example, if the equation is xx+1=5\frac{x}{x+1}=5, you would multiply both sides by (x+1)(x+1) to get x=5(x+1)x=5(x+1).

Q: What if I have a fraction with a variable in the numerator and denominator?

A: If you have a fraction with a variable in the numerator and denominator, you will need to multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fraction. For example, if the equation is x+1x+2=5\frac{x+1}{x+2}=5, you would multiply both sides by (x+2)(x+2) to get x+1=5(x+2)x+1=5(x+2).

Q: How do I solve an equation with multiple fractions?

A: To solve an equation with multiple fractions, you will need to multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions. For example, if the equation is x+12+x+63=5\frac{x+1}{2}+\frac{x+6}{3}=5, you would multiply both sides by 6 to get 3(x+1)+2(x+6)=303(x+1)+2(x+6)=30.

Q: What if I have a fraction with a negative exponent?

A: If you have a fraction with a negative exponent, you will need to multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fraction. For example, if the equation is 1x=5\frac{1}{x}=5, you would multiply both sides by xx to get 1=5x1=5x.

Conclusion

Solving equations involving fractions can be a challenging task, but with the right guidance, it can be a straightforward process. By following the tips and tricks outlined in this article, you can solve equations involving fractions with ease and confidence. Remember to multiply both sides by the least common multiple (LCM) of the denominators, distribute the 6 to each fraction, simplify the fractions, expand and combine like terms, combine like terms further, subtract 15 from both sides, and finally divide both sides by 5 to find the solution. With practice and patience, you can become a master of solving equations involving fractions.

Final Thoughts

Solving equations involving fractions is an essential skill that can be applied to a wide range of mathematical problems. By mastering this skill, you can solve equations involving fractions with ease and confidence. Remember to stay focused, work methodically, and practice regularly to become a master of solving equations involving fractions.