Consider The Steps To Solve The Equation: 2 5 ( 1 2 Y + 20 ) − 4 5 = 9 20 ( 2 Y − 1 \frac{2}{5}\left(\frac{1}{2} Y+20\right)-\frac{4}{5}=\frac{9}{20}(2 Y-1 5 2 ​ ( 2 1 ​ Y + 20 ) − 5 4 ​ = 20 9 ​ ( 2 Y − 1 ]After Distributing, The Equation Becomes: 1 5 Y + 8 − 4 5 = 9 10 Y − 9 20 \frac{1}{5} Y+8-\frac{4}{5}=\frac{9}{10} Y-\frac{9}{20} 5 1 ​ Y + 8 − 5 4 ​ = 10 9 ​ Y − 20 9 ​ What Is The Next Step

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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 explore the steps to solve a linear equation, using the given equation as an example. We will break down the solution into manageable steps, making it easier to understand and follow.

The Given Equation

The given equation is:

25(12y+20)−45=920(2y−1)\frac{2}{5}\left(\frac{1}{2} y+20\right)-\frac{4}{5}=\frac{9}{20}(2 y-1)

This equation appears complex, but with the right steps, we can simplify it and solve for the variable yy.

Step 1: Distribute the Terms

The first step is to distribute the terms inside the parentheses. This involves multiplying each term inside the parentheses by the coefficient outside the parentheses.

25(12y+20)−45=920(2y−1)\frac{2}{5}\left(\frac{1}{2} y+20\right)-\frac{4}{5}=\frac{9}{20}(2 y-1)

25×12y+25×20−45=920×2y−920×1\frac{2}{5} \times \frac{1}{2} y + \frac{2}{5} \times 20 - \frac{4}{5} = \frac{9}{20} \times 2 y - \frac{9}{20} \times 1

Simplifying the equation, we get:

15y+8−45=910y−920\frac{1}{5} y + 8 - \frac{4}{5} = \frac{9}{10} y - \frac{9}{20}

Step 2: Simplify the Equation

The next step is to simplify the equation by combining like terms. This involves combining the constant terms and the variable terms separately.

15y+8−45=910y−920\frac{1}{5} y + 8 - \frac{4}{5} = \frac{9}{10} y - \frac{9}{20}

Combining the constant terms, we get:

8−45=−9208 - \frac{4}{5} = -\frac{9}{20}

Simplifying further, we get:

405−45=−920\frac{40}{5} - \frac{4}{5} = -\frac{9}{20}

365=−920\frac{36}{5} = -\frac{9}{20}

Step 3: Eliminate the Fractions

The next step is to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM is 20.

365=−920\frac{36}{5} = -\frac{9}{20}

Multiplying both sides by 20, we get:

20×365=20×−92020 \times \frac{36}{5} = 20 \times -\frac{9}{20}

Simplifying, we get:

144=−9144 = -9

Step 4: Solve for the Variable

The final step is to solve for the variable yy. However, in this case, we have a contradiction, and the equation has no solution.

144=−9144 = -9

This means that the given equation has no solution, and we cannot find a value for yy that satisfies the equation.

Conclusion

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It is a simple equation that can be solved using basic algebraic operations.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

  1. Distribute the terms inside the parentheses.
  2. Simplify the equation by combining like terms.
  3. Eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
  4. Solve for the variable.

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

A: The least common multiple (LCM) is the smallest multiple that is common to two or more numbers. It is used to eliminate fractions by multiplying both sides of the equation by the LCM.

Q: How do I know if an equation has a solution?

A: To determine if an equation has a solution, you need to check if the equation is true for any value of the variable. If the equation is true for any value of the variable, then it has a solution. If the equation is not true for any value of the variable, then it has no solution.

Q: What is a contradiction in an equation?

A: A contradiction in an equation is a statement that is false. For example, the equation 2=32 = 3 is a contradiction because 2 is not equal to 3.

Q: How do I handle a contradiction in an equation?

A: When you encounter a contradiction in an equation, it means that the equation has no solution. You can conclude that the equation has no solution and move on to the next equation.

Q: Can I solve an equation with multiple variables?

A: Yes, you can solve an equation with multiple variables. To do this, you need to use the same steps as before, but you need to consider all the variables in the equation.

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(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2.

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

A: Yes, you can use a calculator to solve a linear equation. However, it is always a good idea to check your work by hand to make sure that the calculator is giving you the correct answer.

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

A: Some common mistakes to avoid when solving linear equations include:

  • Not distributing the terms inside the parentheses
  • Not simplifying the equation by combining like terms
  • Not eliminating the fractions by multiplying both sides of the equation by the LCM
  • Not checking for contradictions in the equation

By following these steps and avoiding common mistakes, you can become proficient in solving linear equations and move on to more complex equations.