Solve Each Equation.a. 8 X = 2 3 8x = \frac{2}{3} 8 X = 3 2 ​ B. 1 1 2 = 2 X 1 \frac{1}{2} = 2x 1 2 1 ​ = 2 X C. 5 X = 2 7 5x = \frac{2}{7} 5 X = 7 2 ​ D. 1 4 X = 5 \frac{1}{4}x = 5 4 1 ​ X = 5 E. 1 5 = 2 3 X \frac{1}{5} = \frac{2}{3}x 5 1 ​ = 3 2 ​ X

Introduction

Equations are a fundamental concept in mathematics, and solving them is a crucial skill that every student should master. In this article, we will focus on solving a series of equations, each with a unique challenge. We will break down each equation, explain the steps to solve it, and provide examples to illustrate the process.

Equation a: $8x = \frac{2}{3}$

To solve this equation, we need to isolate the variable x. The first step is to get rid of the fraction on the right-hand side. We can do this by multiplying both sides of the equation by the reciprocal of the fraction, which is 3.

Step 1: Multiply both sides by 3

$$8x \times 3 = \frac{2}{3} \times 3$$

This simplifies to:

$$24x = 2$$

Step 2: Divide both sides by 24

$$\frac{24x}{24} = \frac{2}{24}$$

This simplifies to:

$$x = \frac{1}{12}$$

Therefore, the solution to equation a is x = 1/12.

Equation b: $1 \frac{1}{2} = 2x$

To solve this equation, we need to convert the mixed number to an improper fraction. A mixed number is a combination of a whole number and a fraction. In this case, 1 1/2 can be written as 3/2.

Step 1: Convert the mixed number to an improper fraction

$$1 \frac{1}{2} = \frac{3}{2}$$

Step 2: Rewrite the equation with the improper fraction

$$\frac{3}{2} = 2x$$

Step 3: Multiply both sides by 1/2

$$\frac{3}{2} \times \frac{1}{2} = 2x \times \frac{1}{2}$$

This simplifies to:

$$\frac{3}{4} = x$$

Therefore, the solution to equation b is x = 3/4.

Equation c: $5x = \frac{2}{7}$

To solve this equation, we need to isolate the variable x. The first step is to get rid of the fraction on the right-hand side. We can do this by multiplying both sides of the equation by the reciprocal of the fraction, which is 7.

Step 1: Multiply both sides by 7

$$5x \times 7 = \frac{2}{7} \times 7$$

This simplifies to:

$$35x = 2$$

Step 2: Divide both sides by 35

$$\frac{35x}{35} = \frac{2}{35}$$

This simplifies to:

$$x = \frac{2}{35}$$

Therefore, the solution to equation c is x = 2/35.

Equation d: $\frac{1}{4}x = 5$

To solve this equation, we need to isolate the variable x. The first step is to get rid of the fraction on the left-hand side. We can do this by multiplying both sides of the equation by the reciprocal of the fraction, which is 4.

Step 1: Multiply both sides by 4

$$\frac{1}{4}x \times 4 = 5 \times 4$$

This simplifies to:

$$x = 20$$

Therefore, the solution to equation d is x = 20.

Equation e: $\frac{1}{5} = \frac{2}{3}x$

To solve this equation, we need to isolate the variable x. The first step is to get rid of the fraction on the right-hand side. We can do this by multiplying both sides of the equation by the reciprocal of the fraction, which is 3.

Step 1: Multiply both sides by 3

$$\frac{1}{5} \times 3 = \frac{2}{3}x \times 3$$

This simplifies to:

$$\frac{3}{5} = 2x$$

Step 2: Divide both sides by 2

$$\frac{\frac{3}{5}}{2} = \frac{2x}{2}$$

This simplifies to:

$$\frac{3}{10} = x$$

Therefore, the solution to equation e is x = 3/10.

Conclusion

Introduction

In our previous article, we solved a series of equations, each with a unique challenge. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used to solve equations. We will answer common questions, provide examples, and offer tips to help you master the art of solving equations.

Q: What is an equation?

A: An equation is a statement that two expressions are equal. It consists of two parts: the left-hand side (LHS) and the right-hand side (RHS). The LHS and RHS are separated by an equal sign (=).

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. For example, 2x + 3 = 5 is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a linear equation?

A: To solve a linear equation, follow these steps:

1. Simplify the equation by combining like terms.
2. Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
3. Check your solution by plugging it back into the original equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, follow these steps:

1. Factor the equation, if possible.
2. Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
3. Simplify the equation and solve for x.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: How do I simplify a quadratic equation?

A: To simplify a quadratic equation, follow these steps:

1. Factor the equation, if possible.
2. Use the quadratic formula to solve for x.
3. Simplify the equation and solve for x.

Q: What is the difference between a direct and inverse variation?

A: A direct variation is a relationship between two variables in which one variable is directly proportional to the other. For example, y = 2x is a direct variation. An inverse variation, on the other hand, is a relationship between two variables in which one variable is inversely proportional to the other. For example, y = 2/x is an inverse variation.

Q: How do I solve a direct variation equation?

A: To solve a direct variation equation, follow these steps:

1. Write the equation in the form y = kx, where k is the constant of variation.
2. Solve for k by plugging in a value for x and y.
3. Use the value of k to solve for y.

Q: How do I solve an inverse variation equation?

A: To solve an inverse variation equation, follow these steps:

1. Write the equation in the form y = k/x, where k is the constant of variation.
2. Solve for k by plugging in a value for x and y.
3. Use the value of k to solve for y.

Conclusion

Solving equations is a crucial skill that every student should master. In this article, we have provided a Q&A guide to help you understand the concepts and techniques used to solve equations. We have answered common questions, provided examples, and offered tips to help you master the art of solving equations. By following these steps and practicing regularly, you can become proficient in solving equations and tackle even the most challenging problems.