The System of Equations with Infinite Solutions

Oct 13, 2025 by magicalceylontours.com 133 views

Solving Systems of Equations: Infinite Solutions and the Value of $b$

In mathematics, a system of equations is a set of equations that are all true at the same time. Solving a system of equations involves finding the values of the variables that make all the equations true. In this article, we will explore the concept of infinite solutions in systems of equations and determine the value of $b$ in a given system.

Infinite solutions occur when a system of equations has an infinite number of solutions. This happens when the two equations in the system are equivalent, meaning that they represent the same relationship between the variables. In other words, if we can transform one equation into the other by multiplying both sides by the same non-zero constant, then the two equations are equivalent.

Example: Equivalent Equations

Consider the following system of equations:

\begin{array}{c} 2x + 3y = 5 \ 4x + 6y = 10 \end{array}

In this system, the two equations are equivalent because we can transform the first equation into the second equation by multiplying both sides by 2. This means that the two equations represent the same relationship between $x$ and $y$ , and therefore, the system has infinite solutions.

The System of Equations with Infinite Solutions

Now, let's consider the system of equations given in the problem:

\begin{array}{c} 9x - 14y = -3 \ ax - by = 6 \end{array}

To determine the value of $b$ , we need to find the condition under which the two equations are equivalent. This means that we need to find the value of $b$ such that we can transform the first equation into the second equation by multiplying both sides by a non-zero constant.

Transforming the First Equation

Let's start by transforming the first equation into the second equation. We can do this by multiplying both sides of the first equation by a non-zero constant. Let's call this constant $k$ . Then, we have:

k(9x - 14y) = k(-3)

Expanding the left-hand side, we get:

9kx - 14ky = -3k

Now, we want to transform this equation into the second equation. We can do this by multiplying both sides by a non-zero constant. Let's call this constant $m$ . Then, we have:

m(9kx - 14ky) = m(-3k)

Expanding the left-hand side, we get:

9kmx - 14kmy = -3km

Now, we can see that the left-hand side of this equation is equal to the left-hand side of the second equation. Therefore, we have:

9kmx - 14kmy = ax - by

Equating the coefficients of $x$ and $y$ on both sides, we get:

9km = a

-14km = -b

Solving for $b$

Now, we can solve for $b$ by dividing both sides of the second equation by $-14k$ . This gives us:

b = 14km

But we also know that $9km = a$ . Therefore, we can substitute this expression for $a$ into the equation for $b$ . This gives us:

b = 14k(9km)

Simplifying, we get:

b = 126k^2m

But we also know that $-14km = -b$ . Therefore, we can equate the two expressions for $b$ . This gives us:

126k^2m = -b

-b = -b

This is a true statement for all values of $b$ . Therefore, we can conclude that:

b = 126k^2m

In this article, we explored the concept of infinite solutions in systems of equations and determined the value of $b$ in a given system. We showed that the two equations in the system are equivalent when the coefficients of $x$ and $y$ are proportional. We then solved for $b$ by equating the coefficients of $x$ and $y$ on both sides of the equations. Finally, we concluded that the value of $b$ is given by the expression $b = 126k^2m$ .

The final answer is: $\boxed{126}$
Solving Systems of Equations: Infinite Solutions and the Value of $b$ - Q&A

In our previous article, we explored the concept of infinite solutions in systems of equations and determined the value of $b$ in a given system. In this article, we will answer some frequently asked questions (FAQs) related to the topic.

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

A: A system of equations has infinite solutions when the two equations in the system are equivalent. This means that we can transform one equation into the other by multiplying both sides by the same non-zero constant.

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

A: To determine if two equations are equivalent, we need to check if the coefficients of $x$ and $y$ are proportional. If the coefficients are proportional, then the two equations are equivalent.

Q: What is the value of $b$ in the system of equations given in the problem?

A: The value of $b$ is given by the expression $b = 126k^2m$ , where $k$ and $m$ are non-zero constants.

Q: Can we have infinite solutions in a system of equations with more than two equations?

A: Yes, it is possible to have infinite solutions in a system of equations with more than two equations. However, the condition for infinite solutions is that the equations must be linearly dependent, meaning that one equation can be expressed as a linear combination of the other equations.

Q: How do we find the value of $b$ in a system of equations with infinite solutions?

A: To find the value of $b$ in a system of equations with infinite solutions, we need to equate the coefficients of $x$ and $y$ on both sides of the equations. This will give us an expression for $b$ in terms of the other variables.

Q: Can we have infinite solutions in a system of equations with no variables?

A: No, it is not possible to have infinite solutions in a system of equations with no variables. Infinite solutions require the presence of variables, and the equations must be linearly dependent.

Q: How do we check if a system of equations has infinite solutions?

A: To check if a system of equations has infinite solutions, we need to check if the equations are linearly dependent. We can do this by transforming one equation into the other by multiplying both sides by the same non-zero constant.

Q: Can we have infinite solutions in a system of equations with rational coefficients?

A: Yes, it is possible to have infinite solutions in a system of equations with rational coefficients. However, the condition for infinite solutions is that the equations must be linearly dependent, meaning that one equation can be expressed as a linear combination of the other equations.

In this article, we answered some frequently asked questions related to the topic of infinite solutions in systems of equations. We hope that this article has provided you with a better understanding of the concept and how to determine the value of $b$ in a given system.

The final answer is: $\boxed{126}$