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

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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 x3+x−25=6\frac{x}{3} + \frac{x-2}{5} = 6. This type of equation is commonly encountered in mathematics, particularly in algebra and calculus. Our goal is to isolate the variable xx and find its value.

Understanding the Equation

The given equation is a linear equation, which means it can be written in the form ax+b=cax + b = c, where aa, bb, and cc are constants. In this case, the equation is x3+x−25=6\frac{x}{3} + \frac{x-2}{5} = 6. To solve this equation, we need to get rid of the fractions and isolate the variable xx.

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

The first step in solving this equation is to eliminate the fractions. To do this, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is 1515. This will allow us to get rid of the fractions and simplify the equation.

# Import necessary modules
import sympy as sp

x = sp.symbols('x')



equation = sp.Eq(x/3 + (x-2)/5, 6)



equation = sp.Eq(15equation.lhs, 15equation.rhs)

Step 2: Distribute the Multiplication

After multiplying both sides by 1515, we need to distribute the multiplication to simplify the equation. This will give us 5x+3(x−2)=905x + 3(x-2) = 90.

# Distribute the multiplication
equation = sp.Eq(5*x + 3*(x-2), 90)

Step 3: Simplify the Equation

Now that we have distributed the multiplication, we can simplify the equation by combining like terms. This will give us 8x−6=908x - 6 = 90.

# Simplify the equation
equation = sp.Eq(8*x - 6, 90)

Step 4: Add 6 to Both Sides

To isolate the term with the variable xx, we need to add 66 to both sides of the equation. This will give us 8x=968x = 96.

# Add 6 to both sides
equation = sp.Eq(8*x, 96)

Step 5: Divide Both Sides by 8

Finally, we can solve for xx by dividing both sides of the equation by 88. This will give us x=12x = 12.

# Divide both sides by 8
x_value = sp.solve(equation, x)[0]

Conclusion

In this article, we solved the linear equation x3+x−25=6\frac{x}{3} + \frac{x-2}{5} = 6 using a step-by-step approach. We multiplied both sides by the least common multiple (LCM) of the denominators, distributed the multiplication, simplified the equation, added 66 to both sides, and finally divided both sides by 88 to solve for xx. The value of xx is 1212.

Frequently Asked Questions

  • What is the least common multiple (LCM) of the denominators?
    • The LCM of the denominators is 1515.
  • How do I simplify the equation after multiplying both sides by 1515?
    • You can simplify the equation by distributing the multiplication and combining like terms.
  • What is the value of xx?
    • The value of xx is 1212.

References

Further Reading

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Q&A: Solving the Equation

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

A: The LCM of the denominators is 1515.

Q: How do I simplify the equation after multiplying both sides by 1515?

A: You can simplify the equation by distributing the multiplication and combining like terms.

Q: What is the value of xx?

A: The value of xx is 1212.

Q: Can I use a calculator to solve the equation?

A: Yes, you can use a calculator to solve the equation. However, it's always a good idea to understand the steps involved in solving the equation.

Q: What if I have a different equation to solve?

A: The steps involved in solving the equation are the same. You need to multiply both sides by the LCM of the denominators, distribute the multiplication, simplify the equation, add or subtract the same value to both sides, and finally divide both sides by the coefficient of the variable.

Q: Can I use a computer program to solve the equation?

A: Yes, you can use a computer program to solve the equation. Many computer programs, such as Python and MATLAB, have built-in functions to solve linear equations.

Q: What if I have a system of equations to solve?

A: If you have a system of equations, you need to use a different approach to solve it. You can use substitution or elimination methods to solve the system of equations.

Q: Can I use algebraic manipulation to solve the equation?

A: Yes, you can use algebraic manipulation to solve the equation. Algebraic manipulation involves using properties of equations, such as the distributive property and the commutative property, to simplify the equation.

Q: What if I have a quadratic equation to solve?

A: If you have a quadratic equation, you need to use a different approach to solve it. You can use the quadratic formula or factoring to solve the quadratic equation.

Common Mistakes to Avoid

  • Not multiplying both sides of the equation by the LCM of the denominators.
  • Not distributing the multiplication correctly.
  • Not simplifying the equation correctly.
  • Not adding or subtracting the same value to both sides.
  • Not dividing both sides by the coefficient of the variable.

Tips and Tricks

  • Make sure to read the problem carefully and understand what is being asked.
  • Use a step-by-step approach to solve the equation.
  • Check your work by plugging the solution back into the original equation.
  • Use algebraic manipulation to simplify the equation.
  • Use a calculator or computer program to check your work.

Conclusion

Solving linear equations is an important skill in mathematics. By following the steps involved in solving the equation, you can solve a wide range of linear equations. Remember to multiply both sides by the LCM of the denominators, distribute the multiplication, simplify the equation, add or subtract the same value to both sides, and finally divide both sides by the coefficient of the variable. With practice and patience, you can become proficient in solving linear equations.

Frequently Asked Questions

  • What is the least common multiple (LCM) of the denominators?
    • The LCM of the denominators is 1515.
  • How do I simplify the equation after multiplying both sides by 1515?
    • You can simplify the equation by distributing the multiplication and combining like terms.
  • What is the value of xx?
    • The value of xx is 1212.

References

Further Reading