What Value Of $b$ Will Cause The System To Have An Infinite Number Of Solutions?$\[ \begin{align*} y &= 6x + B \\ -3x + \frac{1}{2}y &= -3 \end{align*} \\]

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 are all linear, meaning they can be written in the form of $ax + by = c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables. When we have a system of linear equations, we can use various methods to solve for the values of the variables. However, in this article, we will focus on finding the value of $b$ that will cause the system to have an infinite number of solutions.

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

$$\begin{align*} y &= 6x + b \\ -3x + \frac{1}{2}y &= -3 \end{align*}$$

To find the value of $b$ that will cause the system to have an infinite number of solutions, we need to first solve the system of equations. We can do this by substituting the expression for $y$ from the first equation into the second equation.

Let's substitute the expression for $y$ from the first equation into the second equation:

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

Now, let's simplify the equation by distributing the $\frac{1}{2}$:

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

Next, let's combine like terms:

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

Now, let's solve for $b$ by multiplying both sides of the equation by 2:

$$b = -6$$

So, we have found that the value of $b$ that will cause the system to have an infinite number of solutions is $-6$. But what does this mean? When we have an infinite number of solutions, it means that there are an infinite number of possible values for the variables that satisfy the system of equations.

To understand why this is the case, let's consider the geometric interpretation of the system of equations. The first equation represents a line in the $xy$-plane, and the second equation represents another line in the $xy$-plane. When we have a system of linear equations, we can graph the two lines on the same coordinate plane.

Let's graph the two lines on the same coordinate plane. The first line is given by the equation $y = 6x + b$, and the second line is given by the equation $-3x + \frac{1}{2}y = -3$. We can graph these lines by plotting points on the coordinate plane.

When we graph the two lines on the same coordinate plane, we can see that they intersect at a single point. However, when we substitute the value of $b = -6$ into the first equation, we get the equation $y = 6x - 6$. This equation represents a line that is parallel to the second line.

When we have two parallel lines, they never intersect, and there are an infinite number of points on the coordinate plane that lie on both lines. This is because the two lines have the same slope, but different $y$-intercepts.

In conclusion, we have found that the value of $b$ that will cause the system to have an infinite number of solutions is $-6$. This is because the two lines represented by the system of equations are parallel, and there are an infinite number of points on the coordinate plane that lie on both lines.

In this article, we have explored the concept of infinite solutions in a system of linear equations. We have seen how to find the value of $b$ that will cause the system to have an infinite number of solutions, and we have discussed the geometric interpretation of this concept. We hope that this article has provided a clear and concise explanation of this important mathematical concept.

• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Introduction to Linear Algebra" by Jim Hefferon
• [3] "Linear Algebra: A Modern Introduction" by David Poole
  Frequently Asked Questions: Infinite Solutions in Linear Equations

In our previous article, we explored the concept of infinite solutions in a system of linear equations. We discussed how to find the value of $b$ that will cause the system to have an infinite number of solutions, and we provided a geometric interpretation of this concept. In this article, we will answer some of the most frequently asked questions about infinite solutions in linear equations.

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

A: A system of linear equations with a unique solution has a single solution that satisfies both equations. On the other hand, a system of linear equations with an infinite number of solutions has an infinite number of possible values for the variables that satisfy both equations.

Q: How do I determine if a system of linear equations has an infinite number of solutions?

A: To determine if a system of linear equations has an infinite number of solutions, you need to check if the two equations are parallel. If the two equations are parallel, then the system has an infinite number of solutions.

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 $-6$. This is because the two lines represented by the system of equations are parallel, and there are an infinite number of points on the coordinate plane that lie on both lines.

Q: Can a system of linear equations have both a unique solution and an infinite number of solutions?

A: No, a system of linear equations cannot have both a unique solution and an infinite number of solutions. If a system of linear equations has a unique solution, then it does not have an infinite number of solutions, and vice versa.

Q: How do I graph a system of linear equations with an infinite number of solutions?

A: To graph a system of linear equations with an infinite number of solutions, you need to graph the two lines on the same coordinate plane. The two lines will be parallel, and there will be an infinite number of points on the coordinate plane that lie on both lines.

Q: Can a system of linear equations with an infinite number of solutions be solved using substitution or elimination?

A: No, a system of linear equations with an infinite number of solutions cannot be solved using substitution or elimination. This is because the two equations are parallel, and there is no unique solution to the system.

Q: What is the significance of infinite solutions in linear equations?

A: Infinite solutions in linear equations have significant implications in various fields, including physics, engineering, and economics. For example, in physics, infinite solutions can represent the infinite number of possible paths that a particle can take in a given situation. In engineering, infinite solutions can represent the infinite number of possible designs for a given system. In economics, infinite solutions can represent the infinite number of possible prices for a given good or service.

In conclusion, we have answered some of the most frequently asked questions about infinite solutions in linear equations. We hope that this article has provided a clear and concise explanation of this important mathematical concept.

Infinite solutions in linear equations are an important concept in mathematics that has significant implications in various fields. We hope that this article has provided a useful resource for students and professionals who are interested in learning more about this concept.

• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Introduction to Linear Algebra" by Jim Hefferon
• [3] "Linear Algebra: A Modern Introduction" by David Poole