Solve For { X $} : : : { -\frac{3}{8}x + \frac{1}{6} = -\frac{7}{12} \}

by ADMIN 72 views

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of linear equation, which involves isolating the variable. We will use the given equation as an example and walk through the steps to solve it.

The Given Equation

The given equation is:

−38x+16=−712-\frac{3}{8}x + \frac{1}{6} = -\frac{7}{12}

Understanding the Equation

Before we start solving the equation, let's understand what it means. The equation is in the form of a linear equation, which is a polynomial equation of degree one. The equation has one variable, x, and the coefficients of x and the constant term are fractions.

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. In this case, the denominators are 8, 6, and 12. The LCM of these numbers is 24.

from fractions import Fraction

equation = "-3/8*x + 1/6 = -7/12"



equation = "(-3/8*x + 1/6) * 24 = (-7/12) * 24"


print(equation)

Step 2: Simplify the Equation

After multiplying both sides by the LCM, we can simplify the equation by combining like terms.

−3/8∗x∗24+1/6∗24=−7/12∗24-3/8*x*24 + 1/6*24 = -7/12*24

−9x+4=−14-9x + 4 = -14

Step 3: Isolate the Variable

Now that we have simplified the equation, we can isolate the variable x by subtracting 4 from both sides and then dividing both sides by -9.

−9x=−14−4-9x = -14 - 4

−9x=−18-9x = -18

x=−18/−9x = -18/-9

x=2x = 2

Conclusion

In this article, we solved a linear equation by isolating the variable x. We started by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, then simplified the equation by combining like terms, and finally isolated the variable x by subtracting 4 from both sides and dividing both sides by -9. The final solution is x = 2.

Tips and Tricks

  • When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
  • When multiplying both sides of an equation by a fraction, make sure to multiply both the numerator and the denominator by the fraction.
  • When simplifying an equation, combine like terms to make it easier to isolate the variable.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future economic trends.

Final Thoughts

Solving linear equations is a crucial skill for students to master, and it has numerous real-world applications. By following the steps outlined in this article, you can solve linear equations with ease and confidence. Remember to always follow the order of operations, multiply both sides of the equation by the least common multiple (LCM), simplify the equation by combining like terms, and isolate the variable by subtracting and dividing. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.

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Introduction

In our previous article, we walked through the steps to solve a linear equation by isolating the variable. However, we know that practice makes perfect, and there's no better way to learn than by asking questions and getting answers. In this article, we'll provide a Q&A guide to help you better understand how to solve linear equations.

Q1: What is a linear equation?

A linear equation is a polynomial equation of degree one, which means it has one variable and the coefficients of the variable and the constant term are constants.

A1: Example of a linear equation

The equation x + 2 = 5 is a linear equation because it has one variable (x) and the coefficients of the variable and the constant term are constants.

Q2: How do I know if an equation is linear?

To determine if an equation is linear, look for the following characteristics:

  • The equation has one variable.
  • The coefficients of the variable and the constant term are constants.
  • The equation is in the form of ax + b = c, where a, b, and c are constants.

A2: Example of a non-linear equation

The equation x^2 + 2x + 1 = 0 is not a linear equation because it has a squared variable (x^2), which makes it a quadratic equation.

Q3: What is the least common multiple (LCM)?

The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. In the context of linear equations, the LCM is used to eliminate fractions by multiplying both sides of the equation by the LCM.

A3: Example of finding the LCM

To find the LCM of 8, 6, and 12, we can list the multiples of each number:

  • Multiples of 8: 8, 16, 24, 32, ...
  • Multiples of 6: 6, 12, 18, 24, ...
  • Multiples of 12: 12, 24, 36, 48, ...

The smallest number that appears in all three lists is 24, so the LCM of 8, 6, and 12 is 24.

Q4: How do I simplify an equation?

To simplify an equation, combine like terms by adding or subtracting the coefficients of the same variable.

A4: Example of simplifying an equation

The equation -3x + 4x = 2 can be simplified by combining like terms:

-3x + 4x = 1x 1x = 1

Q5: How do I isolate the variable?

To isolate the variable, subtract the constant term from both sides of the equation and then divide both sides by the coefficient of the variable.

A5: Example of isolating the variable

The equation -9x = -18 can be solved by isolating the variable:

-9x = -18 x = -18/-9 x = 2

Q6: What are some common mistakes to avoid when solving linear equations?

Some common mistakes to avoid when solving linear equations include:

  • Not following the order of operations (PEMDAS)
  • Not multiplying both sides of the equation by the least common multiple (LCM)
  • Not simplifying the equation by combining like terms
  • Not isolating the variable by subtracting and dividing

A6: Tips for avoiding common mistakes

To avoid common mistakes when solving linear equations, make sure to:

  • Follow the order of operations (PEMDAS)
  • Multiply both sides of the equation by the least common multiple (LCM)
  • Simplify the equation by combining like terms
  • Isolate the variable by subtracting and dividing

Conclusion

Solving linear equations is a crucial skill for students to master, and it has numerous real-world applications. By following the steps outlined in this article and practicing with examples, you can become proficient in solving linear equations and apply them to real-world problems. Remember to always follow the order of operations, multiply both sides of the equation by the least common multiple (LCM), simplify the equation by combining like terms, and isolate the variable by subtracting and dividing. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.