Each Equation Below Has A Specific Type Of Solution. Select The Correct Type Of Solution From The Drop-down Menus.a. 10 X − 2 X + 12 = 2 ( 3 X + 5 ) + 2 X 10x - 2x + 12 = 2(3x + 5) + 2x 10 X − 2 X + 12 = 2 ( 3 X + 5 ) + 2 X (Select)b. X + 6 = 1 2 X + 6 X + 6 = \frac{1}{2}x + 6 X + 6 = 2 1 ​ X + 6 (Select)c. 12 X − 5 − 6 X = − 5 + 6 X 12x - 5 - 6x = -5 + 6x 12 X − 5 − 6 X = − 5 + 6 X

Introduction

Linear equations are a fundamental concept in mathematics, and understanding the types of solutions they have is crucial for solving them effectively. In this article, we will explore the different types of solutions that linear equations can have and provide examples to illustrate each type.

What are Linear Equations?

A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form:

ax + b = c

where a, b, and c are constants, and x is the variable.

Types of Solutions

There are three main types of solutions that linear equations can have:

1. Infinite Solutions

An infinite solution occurs when the equation has multiple solutions, and the solutions are not unique. This happens when the equation is an identity, meaning that it is true for all values of the variable.

Example:

a. $10x - 2x + 12 = 2(3x + 5) + 2x$

To solve this equation, we can start by simplifying the left-hand side:

$10x - 2x + 12 = 8x + 12$

Next, we can simplify the right-hand side:

$2(3x + 5) + 2x = 6x + 10 + 2x = 8x + 10$

Now, we can equate the two expressions:

$8x + 12 = 8x + 10$

Subtracting 8x from both sides gives us:

$12 = 10$

This is a contradiction, which means that the equation has no solution. However, if we had started with a different equation, such as:

$10x - 2x + 12 = 2(3x + 5) + 2x$

We would have found that the equation is an identity, and therefore has infinite solutions.

2. Unique Solution

A unique solution occurs when the equation has a single solution, and the solution is unique. This happens when the equation is consistent, meaning that it has a solution that satisfies the equation.

Example:

b. $x + 6 = \frac{1}{2}x + 6$

To solve this equation, we can start by subtracting 6 from both sides:

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

Next, we can multiply both sides by 2 to eliminate the fraction:

$2x = x$

Subtracting x from both sides gives us:

$x = 0$

This is a unique solution, which means that the equation has a single solution.

3. No Solution

A no solution occurs when the equation has no solution, and the equation is inconsistent. This happens when the equation is a contradiction, meaning that it is false for all values of the variable.

Example:

c. $12x - 5 - 6x = -5 + 6x$

To solve this equation, we can start by simplifying the left-hand side:

$12x - 6x - 5 = -5 + 6x$

Next, we can simplify the right-hand side:

$-5 + 6x = -5 + 6x$

Now, we can equate the two expressions:

$6x - 5 = -5 + 6x$

Subtracting 6x from both sides gives us:

$-5 = -5$

This is a contradiction, which means that the equation has no solution.

Conclusion

In conclusion, linear equations can have three main types of solutions: infinite solutions, unique solutions, and no solutions. Understanding the types of solutions that linear equations can have is crucial for solving them effectively. By recognizing the type of solution that an equation has, we can determine the best approach to solving the equation.

Key Takeaways

• Linear equations can have three main types of solutions: infinite solutions, unique solutions, and no solutions.
• Infinite solutions occur when the equation is an identity, meaning that it is true for all values of the variable.
• Unique solutions occur when the equation is consistent, meaning that it has a solution that satisfies the equation.
• No solutions occur when the equation is a contradiction, meaning that it is false for all values of the variable.

Final Thoughts

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form:

ax + b = c

where a, b, and c are constants, and x is the variable.

Q: What are the three main types of solutions that linear equations can have?

A: The three main types of solutions that linear equations can have are:

1. Infinite solutions: An infinite solution occurs when the equation has multiple solutions, and the solutions are not unique. This happens when the equation is an identity, meaning that it is true for all values of the variable.
2. Unique solution: A unique solution occurs when the equation has a single solution, and the solution is unique. This happens when the equation is consistent, meaning that it has a solution that satisfies the equation.
3. No solution: A no solution occurs when the equation has no solution, and the equation is inconsistent. This happens when the equation is a contradiction, meaning that it is false for all values of the variable.

Q: How can I determine the type of solution that a linear equation has?

A: To determine the type of solution that a linear equation has, you can follow these steps:

1. Simplify the equation by combining like terms.
2. Check if the equation is an identity (i.e., true for all values of the variable).
3. Check if the equation is consistent (i.e., has a solution that satisfies the equation).
4. Check if the equation is a contradiction (i.e., false for all values of the variable).

Q: What is an example of an equation with infinite solutions?

A: An example of an equation with infinite solutions is:

10x - 2x + 12 = 2(3x + 5) + 2x

This equation is an identity, meaning that it is true for all values of the variable x.

Q: What is an example of an equation with a unique solution?

A: An example of an equation with a unique solution is:

x + 6 = 1/2x + 6

This equation is consistent, meaning that it has a solution that satisfies the equation.

Q: What is an example of an equation with no solution?

A: An example of an equation with no solution is:

12x - 5 - 6x = -5 + 6x

This equation is a contradiction, meaning that it is false for all values of the variable x.

Q: How can I use linear equations in real-life situations?

A: Linear equations can be used in a variety of real-life situations, such as:

• Modeling population growth or decline
• Calculating the cost of goods or services
• Determining the amount of time it takes to complete a task
• Solving problems involving linear motion or velocity

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

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

• Not simplifying the equation by combining like terms
• Not checking if the equation is an identity, consistent, or a contradiction
• Not using the correct order of operations (PEMDAS)
• Not checking for extraneous solutions

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by:

• Working through example problems in a textbook or online resource
• Creating your own practice problems
• Using online tools or apps to generate practice problems
• Joining a study group or working with a tutor to practice solving linear equations.