What Value Of $b$ Will Cause The System To Have An Infinite Number Of Solutions?$ \begin{array}{l} y = 6x - B \\ -3x + \frac{1}{2}y = -3 \end{array} $A. $b = 2$ B. $b = 4$ C. $b = 6$ D. $b = 8$

What Value of $b$ Will Cause the System to Have an Infinite Number of Solutions?

In mathematics, a system of linear equations is a set of two or more equations that involve two or more variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will explore the conditions under which a system of linear equations will have an infinite number of solutions.

What is an Infinite Number of Solutions?

An infinite number of solutions means that there are an infinite number of values of the variables that satisfy all the equations in the system. In other words, the system has no unique solution, but rather a family of solutions that can be expressed in terms of one or more parameters.

The System of Linear Equations

The system of linear equations we will be working with is:

$$\begin{array}{l} y = 6x - b \\ -3x + \frac{1}{2}y = -3 \end{array}$$

Solving the System of Linear Equations

To solve this system of linear equations, we can use the method of substitution or elimination. Let's use the method of substitution.

First, we can solve the first equation for $y$:

$$y = 6x - b$$

Next, we can substitute this expression for $y$ into the second equation:

$$-3x + \frac{1}{2}(6x - b) = -3$$

Simplifying this equation, we get:

$$-3x + 3x - \frac{1}{2}b = -3$$

Combine like terms:

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

Now, we can solve for $b$:

$$b = 6$$

What Value of $b$ Will Cause the System to Have an Infinite Number of Solutions?

From the previous section, we can see that the system of linear equations has a unique solution when $b = 6$. However, we are interested in finding the value of $b$ that will cause the system to have an infinite number of solutions.

To do this, we need to find the value of $b$ that will make the two equations in the system equivalent. In other words, we need to find the value of $b$ that will make the two equations represent the same line.

Finding the Value of $b$

Let's go back to the first equation:

$$y = 6x - b$$

This equation represents a line with slope $6$ and $y$-intercept $-b$.

Now, let's go back to the second equation:

$$-3x + \frac{1}{2}y = -3$$

We can solve this equation for $y$:

$$y = -6x + 6$$

This equation also represents a line with slope $-6$ and $y$-intercept $6$.

Equating the Two Equations

To find the value of $b$ that will make the two equations equivalent, we need to equate the two equations:

$$6x - b = -6x + 6$$

Simplifying this equation, we get:

$$12x - b = 6$$

Now, we can solve for $b$:

$$b = 12x - 6$$

The Value of $b$

From the previous section, we can see that the value of $b$ that will cause the system to have an infinite number of solutions is:

$$b = 12x - 6$$

However, this is not a specific value of $b$. Instead, it is an expression that involves the variable $x$.

To find the specific value of $b$ that will cause the system to have an infinite number of solutions, we need to find the value of $x$ that will make the expression for $b$ equal to a constant.

Finding the Value of $x$

Let's set the expression for $b$ equal to a constant:

$$12x - 6 = c$$

where $c$ is a constant.

Solving for $x$, we get:

$$x = \frac{c + 6}{12}$$

The Value of $b$

Now that we have found the value of $x$, we can substitute it into the expression for $b$:

$$b = 12x - 6$$

$$b = 12\left(\frac{c + 6}{12}\right) - 6$$

Simplifying this expression, we get:

$$b = c + 6 - 6$$

$$b = c$$

In conclusion, the value of $b$ that will cause the system to have an infinite number of solutions is any value of $b$ that is equal to a constant $c$. In other words, the system will have an infinite number of solutions when $b = c$.

The correct answer is:

• $b = 8$

This is because $b = 8$ is a constant value that will make the expression for $b$ equal to a constant.

The system of linear equations we worked with in this article is a simple example of a system that has an infinite number of solutions. In general, a system of linear equations will have an infinite number of solutions when the two equations in the system are equivalent.

In this case, the two equations are equivalent when $b = 8$. This is because the expression for $b$ is equal to a constant when $b = 8$.

The final answer is $b = 8$.
Q&A: What Value of $b$ Will Cause the System to Have an Infinite Number of Solutions?

Q: What is the condition for a system of linear equations to have an infinite number of solutions?

A: A system of linear equations will have an infinite number of solutions when the two equations in the system are equivalent. This means that the two equations represent the same line.

Q: How can we determine if two equations are equivalent?

A: Two equations are equivalent if they have the same slope and $y$-intercept. In other words, if the two equations represent the same line, then they are equivalent.

Q: How can we find the value of $b$ that will cause the system to have an infinite number of solutions?

A: To find the value of $b$ that will cause the system to have an infinite number of solutions, we need to find the value of $b$ that will make the two equations in the system equivalent. This means that we need to find the value of $b$ that will make the two equations represent the same line.

Q: What is the value of $b$ that will cause the system to have an infinite number of solutions?

A: The value of $b$ that will cause the system to have an infinite number of solutions is any value of $b$ that is equal to a constant $c$. In other words, the system will have an infinite number of solutions when $b = c$.

Q: What is the relationship between the value of $b$ and the value of $x$?

A: The value of $b$ is related to the value of $x$ through the expression $b = 12x - 6$. This means that the value of $b$ depends on the value of $x$.

Q: How can we find the value of $x$ that will make the expression for $b$ equal to a constant?

A: To find the value of $x$ that will make the expression for $b$ equal to a constant, we need to set the expression for $b$ equal to a constant and solve for $x$. This will give us the value of $x$ that will make the expression for $b$ equal to a constant.

Q: What is the value of $b$ that will cause the system to have an infinite number of solutions when $x = 1$?

A: To find the value of $b$ that will cause the system to have an infinite number of solutions when $x = 1$, we need to substitute $x = 1$ into the expression for $b$. This will give us the value of $b$ that will cause the system to have an infinite number of solutions when $x = 1$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $b = 8$. This is because $b = 8$ is a constant value that will make the expression for $b$ equal to a constant.

Q: What is the difference between a system of linear equations with a unique solution and a system with an infinite number of solutions?

A: A system of linear equations with a unique solution has a single solution that satisfies all the equations in the system. A system with an infinite number of solutions has an infinite number of solutions that satisfy all the equations in the system.

Q: How can we determine if a system of linear equations has a unique solution or an infinite number of solutions?

A: We can determine if a system of linear equations has a unique solution or an infinite number of solutions by checking if the two equations in the system are equivalent. If the two equations are equivalent, then the system has an infinite number of solutions. If the two equations are not equivalent, then the system has a unique solution.

Q: What is the relationship between the value of $b$ and the value of $x$ in the system of linear equations?

A: The value of $b$ is related to the value of $x$ through the expression $b = 12x - 6$. This means that the value of $b$ depends on the value of $x$.

Q: How can we find the value of $x$ that will make the expression for $b$ equal to a constant?

A: To find the value of $x$ that will make the expression for $b$ equal to a constant, we need to set the expression for $b$ equal to a constant and solve for $x$. This will give us the value of $x$ that will make the expression for $b$ equal to a constant.

In conclusion, the value of $b$ that will cause the system to have an infinite number of solutions is any value of $b$ that is equal to a constant $c$. In other words, the system will have an infinite number of solutions when $b = c$. The value of $b$ is related to the value of $x$ through the expression $b = 12x - 6$. This means that the value of $b$ depends on the value of $x$.