Which Equation Has Infinitely Many Solutions? Select All That Apply.A. $2(4x - 3) = 3(4x - 3$\]B. $3(4x - 2) = 12x + 6$C. $2(3x - 4) = 6x - 4$D. $3(2x - 4) = 2(3x - 6$\]E. $5(4x - 2) = 20x - 10$

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Which Equation Has Infinitely Many Solutions? Select All That Apply

In mathematics, an equation is a statement that asserts the equality of two mathematical expressions. Equations can be used to solve for unknown values, model real-world situations, and make predictions. However, not all equations have a unique solution. In this article, we will explore which equations have infinitely many solutions.

What Are Infinitely Many Solutions?

Infinitely many solutions occur when an equation is true for all possible values of the variable. In other words, no matter what value we assign to the variable, the equation will always be true. This is in contrast to equations that have a unique solution, where there is only one value that satisfies the equation.

Analyzing the Options

Let's analyze each of the options given:

A. 2(4x−3)=3(4x−3)2(4x - 3) = 3(4x - 3)

To determine if this equation has infinitely many solutions, we can start by expanding both sides of the equation:

2(4x−3)=8x−62(4x - 3) = 8x - 6

3(4x−3)=12x−93(4x - 3) = 12x - 9

Now, we can set the two expressions equal to each other:

8x−6=12x−98x - 6 = 12x - 9

Subtracting 8x8x from both sides gives us:

−6=4x−9-6 = 4x - 9

Adding 99 to both sides gives us:

3=4x3 = 4x

Dividing both sides by 44 gives us:

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

This means that the equation 2(4x−3)=3(4x−3)2(4x - 3) = 3(4x - 3) has a unique solution, which is x=34x = \frac{3}{4}.

B. 3(4x−2)=12x+63(4x - 2) = 12x + 6

To determine if this equation has infinitely many solutions, we can start by expanding both sides of the equation:

3(4x−2)=12x−63(4x - 2) = 12x - 6

12x+6=12x+612x + 6 = 12x + 6

Now, we can see that both sides of the equation are equal, which means that the equation is true for all possible values of xx. Therefore, this equation has infinitely many solutions.

C. 2(3x−4)=6x−42(3x - 4) = 6x - 4

To determine if this equation has infinitely many solutions, we can start by expanding both sides of the equation:

2(3x−4)=6x−82(3x - 4) = 6x - 8

6x−4=6x−46x - 4 = 6x - 4

Now, we can see that both sides of the equation are equal, which means that the equation is true for all possible values of xx. Therefore, this equation has infinitely many solutions.

D. 3(2x−4)=2(3x−6)3(2x - 4) = 2(3x - 6)

To determine if this equation has infinitely many solutions, we can start by expanding both sides of the equation:

3(2x−4)=6x−123(2x - 4) = 6x - 12

2(3x−6)=6x−122(3x - 6) = 6x - 12

Now, we can see that both sides of the equation are equal, which means that the equation is true for all possible values of xx. Therefore, this equation has infinitely many solutions.

E. 5(4x−2)=20x−105(4x - 2) = 20x - 10

To determine if this equation has infinitely many solutions, we can start by expanding both sides of the equation:

5(4x−2)=20x−105(4x - 2) = 20x - 10

20x−10=20x−1020x - 10 = 20x - 10

Now, we can see that both sides of the equation are equal, which means that the equation is true for all possible values of xx. Therefore, this equation has infinitely many solutions.

In conclusion, the equations that have infinitely many solutions are:

  • B. 3(4x−2)=12x+63(4x - 2) = 12x + 6
  • C. 2(3x−4)=6x−42(3x - 4) = 6x - 4
  • D. 3(2x−4)=2(3x−6)3(2x - 4) = 2(3x - 6)
  • E. 5(4x−2)=20x−105(4x - 2) = 20x - 10

Q: What is the difference between an equation with a unique solution and an equation with infinitely many solutions?

A: An equation with a unique solution has only one value that satisfies the equation, whereas an equation with infinitely many solutions has an infinite number of values that satisfy the equation.

Q: How can I determine if an equation has infinitely many solutions?

A: To determine if an equation has infinitely many solutions, you can try to simplify the equation and see if both sides are equal for all possible values of the variable. If both sides are equal, then the equation has infinitely many solutions.

Q: Can an equation have both a unique solution and infinitely many solutions?

A: No, an equation cannot have both a unique solution and infinitely many solutions. If an equation has a unique solution, then it means that there is only one value that satisfies the equation, and therefore it cannot have infinitely many solutions.

Q: Are equations with infinitely many solutions always true?

A: Yes, equations with infinitely many solutions are always true. This means that no matter what value you assign to the variable, the equation will always be true.

Q: Can I use equations with infinitely many solutions to solve real-world problems?

A: Yes, equations with infinitely many solutions can be used to solve real-world problems. For example, if you are trying to find the maximum or minimum value of a function, you can use an equation with infinitely many solutions to find the optimal solution.

Q: How can I apply the concept of infinitely many solutions to other areas of mathematics?

A: The concept of infinitely many solutions can be applied to other areas of mathematics, such as algebra, geometry, and calculus. For example, in algebra, you can use equations with infinitely many solutions to solve systems of linear equations. In geometry, you can use equations with infinitely many solutions to find the equation of a line or a circle.

Q: Are there any limitations to the concept of infinitely many solutions?

A: Yes, there are limitations to the concept of infinitely many solutions. For example, if an equation has infinitely many solutions, it means that the equation is not a function, and therefore it cannot be used to solve for a specific value of the variable.

Q: Can I use technology to help me solve equations with infinitely many solutions?

A: Yes, technology can be used to help you solve equations with infinitely many solutions. For example, you can use a graphing calculator or a computer algebra system to graph the equation and see if it has infinitely many solutions.

Q: How can I practice solving equations with infinitely many solutions?

A: You can practice solving equations with infinitely many solutions by working through examples and exercises in a textbook or online resource. You can also try to come up with your own examples of equations with infinitely many solutions and see if you can solve them.

In conclusion, equations with infinitely many solutions are an important concept in mathematics that can be used to solve real-world problems. By understanding how to determine if an equation has infinitely many solutions, you can apply this concept to other areas of mathematics and solve a wide range of problems.