What Value Of B Will Cause The System To Have An Infinite Number Of Solutions?

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In mathematics, systems of linear equations play a crucial role, and understanding their solutions is fundamental. A system of linear equations can have one solution, no solution, or an infinite number of solutions. The condition for a system to have an infinite number of solutions is particularly interesting and arises when the equations are dependent, meaning they essentially represent the same line. This article delves into the specifics of determining the value of b{ b } that causes the following system of equations to have an infinite number of solutions:

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

This article will explore the steps to solve this problem, providing a comprehensive understanding of the underlying concepts and practical applications. Understanding this concept is essential for students, educators, and anyone involved in fields requiring mathematical analysis.

Understanding Systems of Linear Equations

To address the question, it’s important to first grasp the basics of systems of linear equations. A system of linear equations is a set of two or more linear equations containing the same variables. The solution to a system of linear equations is the set of values for the variables that satisfy all equations simultaneously. Graphically, the solution represents the point(s) where the lines intersect.

Types of Solutions

There are three possible outcomes when solving a system of two linear equations:

  1. One unique solution: The lines intersect at exactly one point. This means there is one set of values for x{ x } and y{ y } that satisfies both equations.
  2. No solution: The lines are parallel and do not intersect. In this case, there are no values for x{ x } and y{ y } that can satisfy both equations simultaneously.
  3. Infinite solutions: The lines are coincident, meaning they are essentially the same line. Any point on the line satisfies both equations, resulting in an infinite number of solutions.

Condition for Infinite Solutions

For a system of two linear equations to have infinite solutions, the equations must be dependent. This means that one equation is a multiple of the other. In simpler terms, if you multiply one equation by a constant, you should be able to obtain the other equation. This condition is crucial in determining the value of b{ b } that results in infinite solutions for the given system.

Analyzing the Given System of Equations

Now, let’s consider the given system of equations:

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

To find the value of b{ b } that leads to infinite solutions, we need to manipulate the equations to see how they can be made identical. The key is to express both equations in the same form, typically the slope-intercept form (y=mx+c{ y = mx + c }) or the standard form (Ax+By=C{ Ax + By = C }).

Transforming the Second Equation

The first equation is already in slope-intercept form. Let's transform the second equation into the same form. The second equation is:

−3x+12y=−3{ -3x + \frac{1}{2}y = -3 }

To isolate y{ y }, we first add 3x{ 3x } to both sides:

12y=3x−3{ \frac{1}{2}y = 3x - 3 }

Next, we multiply both sides by 2 to solve for y{ y }:

y=6x−6{ y = 6x - 6 }

Now, both equations are in slope-intercept form:

\begin{align*}
y &= 6x + b \\
y &= 6x - 6
\end{align*}

Determining the Value of b for Infinite Solutions

For the system to have infinite solutions, the two equations must represent the same line. This means that the slopes and the y-intercepts of the two lines must be equal. Comparing the two equations:

\begin{align*}
y &= 6x + b \\
y &= 6x - 6
\end{align*}

We can see that the slopes are already the same (both are 6). For the lines to be identical, the y-intercepts must also be equal. Therefore, we need to find the value of b{ b } such that:

b=−6{ b = -6 }

Verification

To verify, substitute b=−6{ b = -6 } into the first equation:

y=6x−6{ y = 6x - 6 }

Now we have the system:

\begin{align*}
y &= 6x - 6 \\
y &= 6x - 6
\end{align*}

Both equations are identical, confirming that the system has infinite solutions when b=−6{ b = -6 }.

Graphical Interpretation

The graphical interpretation of this result is straightforward. When b=−6{ b = -6 }, the two equations represent the same line. Any point on this line is a solution to both equations, leading to an infinite number of solutions. If b{ b } were any other value, the lines would either be parallel (no solution) or intersect at a single point (one unique solution).

Visualizing Infinite Solutions

Imagine plotting the two lines on a graph. When b=−6{ b = -6 }, you would only see one line because the two equations overlap perfectly. If b{ b } were, for example, -5, the lines would be parallel and never intersect, indicating no solution. If b{ b } were any other value, the lines would intersect at a single point, representing the unique solution to the system.

Alternative Method: Using Proportionality

Another way to approach this problem is by recognizing that for infinite solutions, the coefficients of x{ x } and y{ y } and the constants must be proportional. Let’s rewrite the equations in the standard form Ax+By=C{ Ax + By = C }:

The first equation y=6x+b{ y = 6x + b } can be rewritten as:

−6x+y=b{ -6x + y = b }

The second equation is already given as:

−3x+12y=−3{ -3x + \frac{1}{2}y = -3 }

For infinite solutions, the ratios of the corresponding coefficients must be equal:

−6−3=112=b−3{ \frac{-6}{-3} = \frac{1}{\frac{1}{2}} = \frac{b}{-3} }

Simplifying the first two ratios:

2=2=b−3{ 2 = 2 = \frac{b}{-3} }

Now, we solve for b{ b }:

2=b−3{ 2 = \frac{b}{-3} }

Multiply both sides by -3:

b=−6{ b = -6 }

This method confirms our earlier result, reinforcing the understanding that b=−6{ b = -6 } is the value that results in infinite solutions.

Practical Applications and Implications

Understanding systems of linear equations and their solutions is not just an academic exercise; it has numerous practical applications in various fields.

Real-World Applications

  1. Economics: In economics, systems of linear equations are used to model supply and demand, market equilibrium, and resource allocation. The condition for infinite solutions might represent a scenario where there is a range of prices and quantities that satisfy both supply and demand equations.
  2. Engineering: Engineers use systems of linear equations to analyze circuits, structural systems, and control systems. Infinite solutions might indicate a state of indeterminacy where multiple configurations satisfy the system's constraints.
  3. Computer Graphics: In computer graphics, linear equations are used to perform transformations such as scaling, rotation, and translation. Understanding the solutions to these equations is crucial for creating realistic and dynamic visual effects.
  4. Linear Programming: Linear programming, a mathematical optimization technique, relies heavily on solving systems of linear equations and inequalities. Identifying conditions for infinite solutions can help in understanding the feasible region and optimizing the objective function.

Implications for Problem Solving

Recognizing the conditions for infinite solutions is crucial for effective problem-solving. When faced with a system of linear equations, it’s essential to:

  1. Check for Dependency: Determine if one equation can be obtained by multiplying the other by a constant. This indicates infinite solutions.
  2. Graphical Analysis: Visualize the equations as lines on a graph. If the lines overlap, there are infinite solutions.
  3. Proportionality: Compare the ratios of coefficients and constants. If the ratios are equal, the system has infinite solutions.

By understanding these implications, students and professionals can efficiently analyze and solve a wide range of problems involving linear systems.

Conclusion

In conclusion, the value of b{ b } that causes the system of equations

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

to have an infinite number of solutions is b=−6{ b = -6 }. This was determined by transforming the equations into slope-intercept form and equating the y-intercepts. Additionally, we verified this result using the concept of proportionality between the coefficients and constants. Understanding the conditions for infinite solutions in systems of linear equations is a fundamental concept with broad applications in mathematics, science, and engineering. This article has provided a comprehensive explanation, ensuring that readers can confidently tackle similar problems and grasp the underlying principles. Whether you are a student learning these concepts for the first time or a professional applying them in your field, the ability to analyze and solve systems of linear equations is an invaluable skill.