Infinite Solutions: The Value Of $b$ Explained

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When we look at systems of linear equations, understanding the nature of their solutions is key. A system can have a unique solution, no solution, or an infinite number of solutions. Today, we're diving deep into how to find the specific value of $b$ that will cause the given system to have an infinite number of solutions. This often happens when one equation is essentially a multiple of the other, meaning they represent the same line. We'll break down the process step-by-step, making it easy to grasp this fundamental concept in algebra. Get ready to explore the intriguing world of linear systems!

Understanding Systems of Linear Equations and Infinite Solutions

A system of linear equations consists of two or more linear equations involving the same set of variables. For instance, the system you provided:

$y=6x+b$ -3x + rac{1}{2}y = -3

represents two lines on a graph. The solution(s) to the system are the points where these lines intersect. A unique solution means the lines intersect at exactly one point. No solution occurs when the lines are parallel and never intersect. An infinite number of solutions arises when the two equations represent the exact same line. In this scenario, every point on the line is a solution to both equations.

To determine when a system has an infinite number of solutions, we need to see if one equation can be transformed into the other through algebraic manipulation. This usually involves multiplying one of the equations by a constant. If, after such manipulation, both equations are identical, then they represent the same line, and thus, there are infinitely many solutions. The crucial part here is identifying the condition that makes these two equations equivalent. Let's look at how to achieve this equivalence for the given system.

Analyzing the Given System of Equations

We are given the following system:

1. $y = 6x + b$
2. -3x + rac{1}{2}y = -3

Our goal is to find the value of $b$ that makes these two equations represent the same line. A common strategy is to rewrite both equations in the same standard form, typically $Ax + By = C$ or $y = mx + c$. The first equation is already in slope-intercept form ($y = mx + c$). Let's rearrange the second equation into slope-intercept form as well.

Starting with the second equation: -3x + rac{1}{2}y = -3

We want to isolate $y$. First, add $3x$ to both sides:

rac{1}{2}y = 3x - 3

Now, multiply both sides by 2 to solve for $y$:

$y = 2(3x - 3)$ $y = 6x - 6$

So, the second equation, when rewritten in slope-intercept form, is $y = 6x - 6$. Now we have our system in a more comparable form:

1. $y = 6x + b$
2. $y = 6x - 6$

Notice that both equations have the same slope, which is 6. This means the lines are either parallel or identical. For the lines to be identical (and thus have an infinite number of solutions), their y-intercepts must also be the same. In the first equation, the y-intercept is $b$. In the second equation, the y-intercept is -6.

Determining the Value of $b$

For the system to have an infinite number of solutions, the two equations must be equivalent. As we've seen, the slope ($m$) for both equations is 6. This means the lines are parallel. For them to be the same line, their y-intercepts must also match.

From equation 1: $y = 6x + b$, the y-intercept is $b$. From equation 2: $y = 6x - 6$, the y-intercept is $-6$.

To have an infinite number of solutions, we must set the y-intercepts equal to each other:

$b = -6$

Therefore, when $b = -6$, the two equations become identical ($y = 6x - 6$), representing the same line. This means every point on this line is a solution to the system, resulting in an infinite number of solutions.

We can also think about this by manipulating one equation to match the other directly. Let's take the second equation, -3x + rac{1}{2}y = -3, and try to make it look like the first equation, $y = 6x + b$.

To get a $y$ term on one side, we can multiply the entire second equation by 2:

2 imes (-3x + rac{1}{2}y) = 2 imes (-3) $-6x + y = -6$

Now, isolate $y$ by adding $6x$ to both sides:

$y = 6x - 6$

Comparing this to the first equation, $y = 6x + b$, we can see that for them to be identical, $b$ must be equal to $-6$. This confirms our previous finding.

Alternative Method: Using Proportions

Another way to determine when a system has infinitely many solutions is by comparing the coefficients of the equations. If a system of two linear equations in the form:

$A_1x + B_1y = C_1$ $A_2x + B_2y = C_2$

has infinitely many solutions, then the coefficients must be proportional. That is:

rac{A_1}{A_2} = rac{B_1}{B_2} = rac{C_1}{C_2}

Let's rewrite our given equations in the standard form $Ax + By = C$:

Equation 1: $y = 6x + b ightarrow -6x + y = b$ Equation 2: -3x + rac{1}{2}y = -3

Now, we can identify the coefficients: $A_1 = -6$, $B_1 = 1$, $C_1 = b$ $A_2 = -3$, B_2 = rac{1}{2}, $C_2 = -3$

For infinite solutions, these ratios must be equal:

rac{-6}{-3} = rac{1}{rac{1}{2}} = rac{b}{-3}

Let's evaluate the first two ratios:

rac{-6}{-3} = 2

rac{1}{rac{1}{2}} = 1 imes rac{2}{1} = 2

Since both the first and second ratios are equal to 2, this confirms that the slopes are the same. Now, we need the third ratio to also be equal to 2:

rac{b}{-3} = 2

To solve for $b$, multiply both sides by -3:

$b = 2 imes (-3)$ $b = -6$

This proportional method also clearly shows that $b$ must be $-6$ for the system to have an infinite number of solutions. This approach is particularly useful when equations are not easily rearranged into slope-intercept form.

Conclusion: The Significance of $b$

In summary, the value of $b$ that causes the given system of linear equations to have an infinite number of solutions is -6. This occurs because, with $b = -6$, both equations simplify to $y = 6x - 6$, meaning they represent the exact same line. When two equations in a system describe the same geometric object (in this case, a line), every point on that object is a valid solution for the system.

Understanding the conditions for unique, no, and infinite solutions is a cornerstone of algebra and is critical for solving more complex mathematical problems. Whether you prefer rewriting equations into slope-intercept form or comparing coefficients using proportions, the outcome remains consistent: $b = -6$ is the key to unlocking infinite solutions for this specific system.

If you'd like to explore more about systems of linear equations and their solutions, the National Council of Teachers of Mathematics (NCTM) offers a wealth of resources and information for educators and students alike. You can find valuable articles and tools on their website at nctm.org.