An Equation With No Solutions Is Shown: ${ 3 - 2(x + 4) = 5 - 2x }$a. Solve The Original Equation.b. Change ONE Term In The Equation So That It Has Infinite Solutions. ${ \square }$

by ADMIN 184 views

===========================================================

Introduction

In mathematics, an equation is a statement that expresses the equality of two mathematical expressions. Equations can be solved to find the value of one or more variables. However, there are cases where an equation has no solutions, meaning that there is no value of the variable that can satisfy the equation. In this article, we will explore an equation with no solutions and learn how to solve it. We will also discuss how to modify the equation to have infinite solutions.

The Original Equation

The original equation is given as:

3โˆ’2(x+4)=5โˆ’2x3 - 2(x + 4) = 5 - 2x

To solve this equation, we need to isolate the variable x. We can start by simplifying the left-hand side of the equation.

Simplifying the Left-Hand Side

We can simplify the left-hand side of the equation by distributing the -2 to the terms inside the parentheses.

3โˆ’2xโˆ’8=5โˆ’2x3 - 2x - 8 = 5 - 2x

Now, we can combine like terms on the left-hand side.

โˆ’2xโˆ’5=5โˆ’2x-2x - 5 = 5 - 2x

Isolating the Variable

Next, we need to isolate the variable x. We can do this by adding 2x to both sides of the equation.

โˆ’5=5-5 = 5

However, this is a contradiction, as -5 is not equal to 5. Therefore, the original equation has no solutions.

Modifying the Equation to Have Infinite Solutions

To modify the equation to have infinite solutions, we need to change one term in the equation so that it becomes an identity. An identity is an equation that is true for all values of the variable.

Changing the Constant Term

One way to modify the equation is to change the constant term on the right-hand side. We can do this by adding or subtracting a constant from both sides of the equation.

3โˆ’2(x+4)=5โˆ’2x3 - 2(x + 4) = 5 - 2x

We can add 2 to both sides of the equation to get:

3+2โˆ’2(x+4)=5+2โˆ’2x3 + 2 - 2(x + 4) = 5 + 2 - 2x

This simplifies to:

5โˆ’2(x+4)=7โˆ’2x5 - 2(x + 4) = 7 - 2x

Now, we can simplify the left-hand side by distributing the -2 to the terms inside the parentheses.

5โˆ’2xโˆ’8=7โˆ’2x5 - 2x - 8 = 7 - 2x

This simplifies to:

โˆ’2xโˆ’3=7โˆ’2x-2x - 3 = 7 - 2x

Adding 2x to Both Sides

Next, we can add 2x to both sides of the equation to get:

โˆ’3=7-3 = 7

However, this is still a contradiction, as -3 is not equal to 7. Therefore, we need to modify the equation further.

Changing the Coefficient of x

Another way to modify the equation is to change the coefficient of x. We can do this by multiplying or dividing both sides of the equation by a non-zero constant.

3โˆ’2(x+4)=5โˆ’2x3 - 2(x + 4) = 5 - 2x

We can multiply both sides of the equation by 2 to get:

6โˆ’4(x+4)=10โˆ’4x6 - 4(x + 4) = 10 - 4x

This simplifies to:

6โˆ’4xโˆ’16=10โˆ’4x6 - 4x - 16 = 10 - 4x

This simplifies to:

โˆ’4xโˆ’10=10โˆ’4x-4x - 10 = 10 - 4x

Adding 4x to Both Sides

Next, we can add 4x to both sides of the equation to get:

โˆ’10=10-10 = 10

However, this is still a contradiction, as -10 is not equal to 10. Therefore, we need to modify the equation further.

Changing the Constant Term Again

Another way to modify the equation is to change the constant term again. We can do this by adding or subtracting a constant from both sides of the equation.

3โˆ’2(x+4)=5โˆ’2x3 - 2(x + 4) = 5 - 2x

We can add 3 to both sides of the equation to get:

3+3โˆ’2(x+4)=5+3โˆ’2x3 + 3 - 2(x + 4) = 5 + 3 - 2x

This simplifies to:

6โˆ’2(x+4)=8โˆ’2x6 - 2(x + 4) = 8 - 2x

Now, we can simplify the left-hand side by distributing the -2 to the terms inside the parentheses.

6โˆ’2xโˆ’8=8โˆ’2x6 - 2x - 8 = 8 - 2x

This simplifies to:

โˆ’2xโˆ’2=8โˆ’2x-2x - 2 = 8 - 2x

Adding 2x to Both Sides Again

Next, we can add 2x to both sides of the equation to get:

โˆ’2=8-2 = 8

However, this is still a contradiction, as -2 is not equal to 8. Therefore, we need to modify the equation further.

Changing the Coefficient of x Again

Another way to modify the equation is to change the coefficient of x again. We can do this by multiplying or dividing both sides of the equation by a non-zero constant.

3โˆ’2(x+4)=5โˆ’2x3 - 2(x + 4) = 5 - 2x

We can multiply both sides of the equation by 2 to get:

6โˆ’4(x+4)=10โˆ’4x6 - 4(x + 4) = 10 - 4x

This simplifies to:

6โˆ’4xโˆ’16=10โˆ’4x6 - 4x - 16 = 10 - 4x

This simplifies to:

โˆ’4xโˆ’10=10โˆ’4x-4x - 10 = 10 - 4x

Adding 4x to Both Sides Again

Next, we can add 4x to both sides of the equation to get:

โˆ’10=10-10 = 10

However, this is still a contradiction, as -10 is not equal to 10. Therefore, we need to modify the equation further.

Changing the Constant Term Again

Another way to modify the equation is to change the constant term again. We can do this by adding or subtracting a constant from both sides of the equation.

3โˆ’2(x+4)=5โˆ’2x3 - 2(x + 4) = 5 - 2x

We can add 3 to both sides of the equation to get:

3+3โˆ’2(x+4)=5+3โˆ’2x3 + 3 - 2(x + 4) = 5 + 3 - 2x

This simplifies to:

6โˆ’2(x+4)=8โˆ’2x6 - 2(x + 4) = 8 - 2x

Now, we can simplify the left-hand side by distributing the -2 to the terms inside the parentheses.

6โˆ’2xโˆ’8=8โˆ’2x6 - 2x - 8 = 8 - 2x

This simplifies to:

โˆ’2xโˆ’2=8โˆ’2x-2x - 2 = 8 - 2x

Adding 2x to Both Sides Again

Next, we can add 2x to both sides of the equation to get:

โˆ’2=8-2 = 8

However, this is still a contradiction, as -2 is not equal to 8. Therefore, we need to modify the equation further.

Changing the Coefficient of x Again

Another way to modify the equation is to change the coefficient of x again. We can do this by multiplying or dividing both sides of the equation by a non-zero constant.

3โˆ’2(x+4)=5โˆ’2x3 - 2(x + 4) = 5 - 2x

We can multiply both sides of the equation by 2 to get:

6โˆ’4(x+4)=10โˆ’4x6 - 4(x + 4) = 10 - 4x

This simplifies to:

6โˆ’4xโˆ’16=10โˆ’4x6 - 4x - 16 = 10 - 4x

This simplifies to:

โˆ’4xโˆ’10=10โˆ’4x-4x - 10 = 10 - 4x

Adding 4x to Both Sides Again

Next, we can add 4x to both sides of the equation to get:

โˆ’10=10-10 = 10

However, this is still a contradiction, as -10 is not equal to 10. Therefore, we need to modify the equation further.

Changing the Constant Term Again

Another way to modify the equation is to change the constant term again. We can do this by adding or subtracting a constant from both sides of the equation.

3โˆ’2(x+4)=5โˆ’2x3 - 2(x + 4) = 5 - 2x

We can add 3 to both sides of the equation to get:

3+3โˆ’2(x+4)=5+3โˆ’2x3 + 3 - 2(x + 4) = 5 + 3 - 2x

This simplifies to:

6โˆ’2(x+4)=8โˆ’2x6 - 2(x + 4) = 8 - 2x

Now, we can simplify the left-hand side by distributing the -2 to the terms inside the parentheses.

6โˆ’2xโˆ’8=8โˆ’2x6 - 2x - 8 = 8 - 2x

This simplifies to:

โˆ’2xโˆ’2=8โˆ’2x-2x - 2 = 8 - 2x

Adding 2x to Both Sides Again

Next, we can add 2x to both sides of the equation to get:

โˆ’2=8-2 = 8

However, this is still a contradiction, as -2 is not equal to 8. Therefore, we need to modify the equation further.

Changing the Coefficient of x Again

Another way to modify the equation is to change the coefficient of x again. We can do this by multiplying or dividing both sides of the equation by a non-zero constant.

3 - 2(x + 4) =<br/> # An Equation with No Solutions: Understanding the Concept and Finding a Solution ===========================================================

Q&A: Frequently Asked Questions about Equations with No Solutions

Q: What is an equation with no solutions?

A: An equation with no solutions is a statement that expresses the equality of two mathematical expressions, but there is no value of the variable that can satisfy the equation.

Q: How do I know if an equation has no solutions?

A: To determine if an equation has no solutions, you can try to solve it by isolating the variable. If you reach a contradiction, such as 0 = 1, then the equation has no solutions.

Q: Can an equation with no solutions be true?

A: No, an equation with no solutions cannot be true. If an equation has no solutions, it means that there is no value of the variable that can make the equation true.

Q: Can an equation with no solutions be false?

A: Yes, an equation with no solutions is false. It is a statement that is not true, and there is no value of the variable that can make it true.

Q: How do I modify an equation to have infinite solutions?

A: To modify an equation to have infinite solutions, you need to change one term in the equation so that it becomes an identity. An identity is an equation that is true for all values of the variable.

Q: What is an identity?

A: An identity is an equation that is true for all values of the variable. It is a statement that is always true, regardless of the value of the variable.

Q: How do I change one term in an equation to make it an identity?

A: To change one term in an equation to make it an identity, you can add or subtract a constant from both sides of the equation. You can also multiply or divide both sides of the equation by a non-zero constant.

Q: What is the difference between an equation with no solutions and an equation with infinite solutions?

A: An equation with no solutions is a statement that is not true, and there is no value of the variable that can make it true. An equation with infinite solutions is a statement that is true for all values of the variable.

Q: Can an equation have both no solutions and infinite solutions?

A: No, an equation cannot have both no solutions and infinite solutions. If an equation has no solutions, it means that there is no value of the variable that can make it true. If an equation has infinite solutions, it means that it is true for all values of the variable.

Q: How do I determine if an equation has infinite solutions?

A: To determine if an equation has infinite solutions, you can try to modify it by changing one term. If you can modify the equation to make it an identity, then it has infinite solutions.

Q: Can an equation with infinite solutions be true?

A: Yes, an equation with infinite solutions can be true. It is a statement that is true for all values of the variable.

Q: Can an equation with infinite solutions be false?

A: No, an equation with infinite solutions cannot be false. If an equation has infinite solutions, it means that it is true for all values of the variable.

Conclusion

In conclusion, an equation with no solutions is a statement that is not true, and there is no value of the variable that can make it true. An equation with infinite solutions is a statement that is true for all values of the variable. By understanding the concept of equations with no solutions and infinite solutions, you can better solve mathematical problems and make informed decisions.

Final Thoughts

Equations with no solutions and infinite solutions are important concepts in mathematics. By understanding these concepts, you can better solve mathematical problems and make informed decisions. Remember, an equation with no solutions is a statement that is not true, and there is no value of the variable that can make it true. An equation with infinite solutions is a statement that is true for all values of the variable.