Solve The Equation.$\[ \frac{1}{6}d + \frac{2}{3} = \frac{1}{4}(d - 2) \\]\[$d = \$\]

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Introduction

In this article, we will delve into solving a linear equation involving fractions. The equation is given as 16d+23=14(dβˆ’2)\frac{1}{6}d + \frac{2}{3} = \frac{1}{4}(d - 2). Our goal is to isolate the variable dd and find its value. We will use algebraic techniques to simplify the equation and solve for dd.

Understanding the Equation

The given equation is a linear equation, which means it can be written in the form ax+b=cx+dax + b = cx + d, where aa, bb, cc, and dd are constants. In this case, the equation is 16d+23=14(dβˆ’2)\frac{1}{6}d + \frac{2}{3} = \frac{1}{4}(d - 2). To solve for dd, we need to isolate the variable on one side of the equation.

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

The first step in solving the equation is to eliminate the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 6 and 4 is 12.

\frac{1}{6}d + \frac{2}{3} = \frac{1}{4}(d - 2)
\\]
12 \times \left(\frac{1}{6}d + \frac{2}{3}\right) = 12 \times \left(\frac{1}{4}(d - 2)\right)
\\]
2d + 8 = 3(d - 2)

Step 2: Distribute the 3 on the Right-Hand Side

Next, we need to distribute the 3 on the right-hand side of the equation.

2d + 8 = 3d - 6

Step 3: Subtract 2d from Both Sides

Now, we need to isolate the variable dd on one side of the equation. We can do this by subtracting 2d from both sides of the equation.

2d - 2d + 8 = 3d - 2d - 6
\\]
8 = d - 6

Step 4: Add 6 to Both Sides

Finally, we need to add 6 to both sides of the equation to solve for dd.

8 + 6 = d - 6 + 6
\\]
14 = d

Conclusion

In this article, we solved the linear equation 16d+23=14(dβˆ’2)\frac{1}{6}d + \frac{2}{3} = \frac{1}{4}(d - 2) using algebraic techniques. We multiplied both sides of the equation by the LCM of the denominators, distributed the 3 on the right-hand side, subtracted 2d from both sides, and added 6 to both sides to solve for dd. The final solution is d=14d = 14.

Tips and Tricks

  • When solving linear equations involving fractions, it's essential to eliminate the fractions by multiplying both sides of the equation by the LCM of the denominators.
  • Distributing the coefficients on the right-hand side of the equation can help simplify the equation and make it easier to solve.
  • Subtracting the same value from both sides of the equation can help isolate the variable on one side of the equation.
  • Adding the same value to both sides of the equation can help solve for the variable.

Frequently Asked Questions

  • What is the least common multiple (LCM) of 6 and 4?
    • The LCM of 6 and 4 is 12.
  • How do I eliminate fractions in a linear equation?
    • You can eliminate fractions by multiplying both sides of the equation by the LCM of the denominators.
  • What is the final solution to the equation 16d+23=14(dβˆ’2)\frac{1}{6}d + \frac{2}{3} = \frac{1}{4}(d - 2)?
    • The final solution is d=14d = 14.

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Introduction

In our previous article, we solved the linear equation 16d+23=14(dβˆ’2)\frac{1}{6}d + \frac{2}{3} = \frac{1}{4}(d - 2) using algebraic techniques. In this article, we will answer some frequently asked questions related to solving linear equations.

Q&A

Q: What is the least common multiple (LCM) of 6 and 4?

A: The LCM of 6 and 4 is 12.

Q: How do I eliminate fractions in a linear equation?

A: You can eliminate fractions by multiplying both sides of the equation by the LCM of the denominators.

Q: What is the final solution to the equation 16d+23=14(dβˆ’2)\frac{1}{6}d + \frac{2}{3} = \frac{1}{4}(d - 2)?

A: The final solution is d=14d = 14.

Q: How do I distribute the coefficients on the right-hand side of the equation?

A: To distribute the coefficients, multiply each term inside the parentheses by the coefficient outside the parentheses.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a linear equation with multiple variables?

A: To solve a linear equation with multiple variables, you can use the method of substitution or elimination. The method of substitution involves substituting one variable in terms of another variable, while the method of elimination involves adding or subtracting the equations to eliminate one variable.

Q: What is the importance of solving linear equations?

A: Solving linear equations is important in many real-world applications, such as physics, engineering, and economics. It helps us to model and analyze complex systems, make predictions, and make informed decisions.

Tips and Tricks

  • When solving linear equations, it's essential to eliminate fractions by multiplying both sides of the equation by the LCM of the denominators.
  • Distributing the coefficients on the right-hand side of the equation can help simplify the equation and make it easier to solve.
  • Subtracting the same value from both sides of the equation can help isolate the variable on one side of the equation.
  • Adding the same value to both sides of the equation can help solve for the variable.

Conclusion

In this article, we answered some frequently asked questions related to solving linear equations. We discussed the importance of eliminating fractions, distributing coefficients, and using the method of substitution or elimination to solve linear equations with multiple variables. We also highlighted the importance of solving linear equations in real-world applications.

Further Reading

  • For more information on solving linear equations, check out our previous article on solving the equation 16d+23=14(dβˆ’2)\frac{1}{6}d + \frac{2}{3} = \frac{1}{4}(d - 2).
  • For more information on linear equations and quadratic equations, check out our article on the difference between linear and quadratic equations.
  • For more information on solving linear equations with multiple variables, check out our article on the method of substitution and elimination.

Resources

  • Khan Academy: Solving Linear Equations
  • Mathway: Solving Linear Equations
  • Wolfram Alpha: Solving Linear Equations

Final Thoughts

Solving linear equations is an essential skill in mathematics and has many real-world applications. By understanding how to eliminate fractions, distribute coefficients, and use the method of substitution or elimination, you can solve linear equations with ease. Remember to practice regularly and seek help when needed. With patience and persistence, you can master the art of solving linear equations.