A Teacher Wrote The Equation 3 Y + 12 = 6 X 3y + 12 = 6x 3 Y + 12 = 6 X On The Board. For What Value Of B B B Would The Additional Equation 2 Y = 4 X + B 2y = 4x + B 2 Y = 4 X + B Form A System Of Linear Equations With Infinitely Many Solutions?A. B = − 8 B = -8 B = − 8 B. $b =

Introduction

In mathematics, a system of linear equations is a set of two or more equations that involve variables raised to the power of 1. These equations can be solved using various methods, including substitution and elimination. However, there are certain conditions under which a system of linear equations will have infinitely many solutions. In this article, we will explore one such condition and determine the value of $b$ that would make the additional equation $2y = 4x + b$ form a system of linear equations with infinitely many solutions.

What are Infinitely Many Solutions?

Infinitely many solutions occur when two linear equations represent the same line. In other words, the two equations are equivalent and have the same slope and y-intercept. This means that any point on one line is also a point on the other line, resulting in an infinite number of solutions.

The Condition for Infinitely Many Solutions

For a system of linear equations to have infinitely many solutions, the two equations must be equivalent. This can be achieved when the two equations have the same slope and y-intercept. In the case of the given equations $3y + 12 = 6x$ and $2y = 4x + b$, we need to find the value of $b$ that makes the two equations equivalent.

Solving the First Equation for y

To begin, let's solve the first equation for $y$.

$$3y + 12 = 6x$$

Subtracting 12 from both sides gives:

$$3y = 6x - 12$$

Dividing both sides by 3 gives:

$$y = 2x - 4$$

Equating the Two Equations

Now that we have solved the first equation for $y$, we can equate it with the second equation.

$$y = 2x - 4$$

$$2y = 4x + b$$

Substituting $y = 2x - 4$ into the second equation gives:

$$2(2x - 4) = 4x + b$$

Expanding the left-hand side gives:

$$4x - 8 = 4x + b$$

Solving for b

Now that we have equated the two equations, we can solve for $b$. Subtracting $4x$ from both sides gives:

$$-8 = b$$

However, this is not the correct solution. We need to find the value of $b$ that makes the two equations equivalent. Let's re-examine the equation $2y = 4x + b$.

Rearranging the Second Equation

Rearranging the second equation to isolate $y$ gives:

$$y = \frac{4x + b}{2}$$

Equating the Two Equations Again

Now that we have rearranged the second equation, we can equate it with the first equation.

$$y = 2x - 4$$

$$y = \frac{4x + b}{2}$$

Equating the two equations gives:

$$2x - 4 = \frac{4x + b}{2}$$

Multiplying both sides by 2 gives:

$$4x - 8 = 4x + b$$

Subtracting $4x$ from both sides gives:

$$-8 = b$$

However, this is still not the correct solution. We need to find the value of $b$ that makes the two equations equivalent. Let's re-examine the equation $2y = 4x + b$.

Finding the Value of b

To find the value of $b$, we need to find the value that makes the two equations equivalent. This can be achieved when the two equations have the same slope and y-intercept. In the case of the given equations $3y + 12 = 6x$ and $2y = 4x + b$, we need to find the value of $b$ that makes the two equations equivalent.

Solving the Second Equation for y

To begin, let's solve the second equation for $y$.

$$2y = 4x + b$$

Dividing both sides by 2 gives:

$$y = 2x + \frac{b}{2}$$

Equating the Two Equations Again

Now that we have solved the second equation for $y$, we can equate it with the first equation.

$$y = 2x - 4$$

$$y = 2x + \frac{b}{2}$$

Equating the two equations gives:

$$2x - 4 = 2x + \frac{b}{2}$$

Subtracting $2x$ from both sides gives:

$$-4 = \frac{b}{2}$$

Multiplying both sides by 2 gives:

$$-8 = b$$

Conclusion

In conclusion, the value of $b$ that would make the additional equation $2y = 4x + b$ form a system of linear equations with infinitely many solutions is $-8$. This is because the two equations have the same slope and y-intercept, resulting in an infinite number of solutions.

Final Answer

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Introduction

In our previous article, we explored the condition for a system of linear equations to have infinitely many solutions. We found that the two equations must be equivalent, meaning they have the same slope and y-intercept. In this article, we will answer some frequently asked questions about systems of linear equations with infinitely many solutions.

Q: What is the difference between a system of linear equations with infinitely many solutions and a system with no solution?

A: A system of linear equations with infinitely many solutions has two equations that represent the same line. This means that any point on one line is also a point on the other line, resulting in an infinite number of solutions. On the other hand, a system with no solution has two equations that are parallel and never intersect.

Q: How can I determine if a system of linear equations has infinitely many solutions?

A: To determine if a system of linear equations has infinitely many solutions, you can check if the two equations are equivalent. This can be done by comparing the slopes and y-intercepts of the two equations. If the slopes and y-intercepts are the same, then the two equations are equivalent and the system has infinitely many solutions.

Q: What is the value of b that makes the additional equation 2y = 4x + b form a system of linear equations with infinitely many solutions?

A: The value of b that makes the additional equation 2y = 4x + b form a system of linear equations with infinitely many solutions is -8. This is because the two equations have the same slope and y-intercept, resulting in an infinite number of solutions.

Q: Can a system of linear equations have infinitely many solutions if the two equations are not equivalent?

A: No, a system of linear equations cannot have infinitely many solutions if the two equations are not equivalent. If the two equations are not equivalent, then they represent different lines and will never intersect, resulting in no solution.

Q: How can I find the value of b that makes the additional equation 2y = 4x + b form a system of linear equations with infinitely many solutions?

A: To find the value of b that makes the additional equation 2y = 4x + b form a system of linear equations with infinitely many solutions, you can follow these steps:

1. Solve the first equation for y.
2. Equate the two equations.
3. Solve for b.

By following these steps, you can find the value of b that makes the additional equation 2y = 4x + b form a system of linear equations with infinitely many solutions.

Q: What is the significance of a system of linear equations with infinitely many solutions?

A: A system of linear equations with infinitely many solutions is significant because it represents a situation where there are an infinite number of possible solutions. This can be useful in real-world applications, such as finding the maximum or minimum value of a function.

Conclusion

In conclusion, a system of linear equations with infinitely many solutions is a situation where two equations represent the same line. This can be determined by checking if the two equations are equivalent. The value of b that makes the additional equation 2y = 4x + b form a system of linear equations with infinitely many solutions is -8. We hope this article has provided you with a better understanding of systems of linear equations with infinitely many solutions.

Final Answer

The final answer is -8.