Home Work 1. Find Value Of K Such That Equations -x + 3y - 3 = 0 And -3x+ky - 9 = 0, Represents Coincident Lines. 2. For What Value Of K, The Following Pair Of Linear Equations Has Infinite Number Of Solutions? Kx+3y=(k+1)- 20k+1x+9v=(7k+1)
Solving Linear Equations: Finding the Value of k for Coincident Lines and Infinite Solutions
In this article, we will explore the concept of linear equations and how to find the value of k that makes two linear equations represent coincident lines or have an infinite number of solutions. We will start by understanding the basics of linear equations and then move on to the specific problems given in the homework.
What are Linear Equations?
A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables. Linear equations can be represented graphically as a straight line on a coordinate plane.
Coincident Lines
Two lines are said to be coincident if they have the same equation and therefore represent the same line. In other words, if two linear equations have the same slope and y-intercept, they are coincident.
Problem 1: Finding the Value of k for Coincident Lines
The given linear equations are:
- -x + 3y - 3 = 0
- -3x + ky - 9 = 0
To find the value of k that makes these two equations represent coincident lines, we need to compare their slopes and y-intercepts.
First, let's rewrite the equations in the slope-intercept form (y = mx + c), where m is the slope and c is the y-intercept.
For the first equation:
y = (1/3)x + 1
For the second equation:
y = (k/3)x + 3
Since the two lines are coincident, their slopes and y-intercepts must be equal. Therefore, we can set up the following equations:
m1 = m2 (1/3) = (k/3) c1 = c2 1 = 3
From the first equation, we can solve for k:
(1/3) = (k/3) 1 = k
So, the value of k that makes the two equations represent coincident lines is k = 1.
Infinite Solutions
A system of linear equations has an infinite number of solutions if the two equations represent the same line. In other words, if the two equations have the same slope and y-intercept, they have an infinite number of solutions.
Problem 2: Finding the Value of k for Infinite Solutions
The given linear equations are:
kx + 3y = (k+1) - 20 (k+1)x + 9y = (7k+1)
To find the value of k that makes these two equations have an infinite number of solutions, we need to compare their slopes and y-intercepts.
First, let's rewrite the equations in the slope-intercept form (y = mx + c), where m is the slope and c is the y-intercept.
For the first equation:
y = (-k/3)x + (k+1)/3 - 20/3
For the second equation:
y = (-k-1)/9)x + (7k+1)/9
Since the two lines have an infinite number of solutions, their slopes and y-intercepts must be equal. Therefore, we can set up the following equations:
m1 = m2 (-k/3) = (-k-1)/9 c1 = c2 ((k+1)/3 - 20/3) = (7k+1)/9
From the first equation, we can solve for k:
(-k/3) = (-k-1)/9 -9k = -3k - 3 -6k = -3 k = 1/2
So, the value of k that makes the two equations have an infinite number of solutions is k = 1/2.
In this article, we have explored the concept of linear equations and how to find the value of k that makes two linear equations represent coincident lines or have an infinite number of solutions. We have used the slope-intercept form of linear equations to compare the slopes and y-intercepts of the two equations and find the value of k.
Key Takeaways
- Two linear equations represent coincident lines if they have the same slope and y-intercept.
- A system of linear equations has an infinite number of solutions if the two equations represent the same line.
- To find the value of k that makes two linear equations represent coincident lines or have an infinite number of solutions, we need to compare their slopes and y-intercepts.
References
Further Reading
- Linear Algebra
- Geometry
- Algebra
Solving Linear Equations: Q&A
In our previous article, we explored the concept of linear equations and how to find the value of k that makes two linear equations represent coincident lines or have an infinite number of solutions. In this article, we will answer some frequently asked questions related to linear equations and provide additional examples to help you understand the concepts better.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1, whereas a quadratic equation is an equation in which the highest power of the variable(s) is 2.
Example:
Linear Equation: 2x + 3y = 5
Quadratic Equation: x^2 + 4x + 4 = 0
Q: How do I determine if two linear equations are coincident or parallel?
A: To determine if two linear equations are coincident or parallel, you need to compare their slopes and y-intercepts. If the slopes and y-intercepts are equal, the lines are coincident. If the slopes are equal but the y-intercepts are different, the lines are parallel.
Example:
Equation 1: y = 2x + 1
Equation 2: y = 2x + 3
In this example, the slopes are equal (2), but the y-intercepts are different (1 and 3). Therefore, the lines are parallel.
Q: What is the significance of the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation (y = mx + c) is significant because it allows us to easily compare the slopes and y-intercepts of two linear equations. It also makes it easy to graph the equation on a coordinate plane.
Example:
Equation: y = 2x + 3
In this example, the slope is 2 and the y-intercept is 3. This information can be used to graph the equation on a coordinate plane.
Q: How do I find the value of k that makes two linear equations have an infinite number of solutions?
A: To find the value of k that makes two linear equations have an infinite number of solutions, you need to compare their slopes and y-intercepts. If the slopes and y-intercepts are equal, the lines have an infinite number of solutions.
Example:
Equation 1: kx + 3y = (k+1) - 20
Equation 2: (k+1)x + 9y = (7k+1)
In this example, we need to find the value of k that makes the two equations have an infinite number of solutions. To do this, we can compare their slopes and y-intercepts.
Q: What is the significance of the concept of infinite solutions in linear equations?
A: The concept of infinite solutions in linear equations is significant because it means that the two equations represent the same line. This can be useful in solving systems of linear equations.
Example:
Equation 1: x + 2y = 3
Equation 2: 2x + 4y = 6
In this example, the two equations have an infinite number of solutions because they represent the same line.
In this article, we have answered some frequently asked questions related to linear equations and provided additional examples to help you understand the concepts better. We have also discussed the significance of the slope-intercept form of a linear equation and the concept of infinite solutions in linear equations.
Key Takeaways
- A linear equation is an equation in which the highest power of the variable(s) is 1.
- Two linear equations are coincident if they have the same slope and y-intercept.
- A system of linear equations has an infinite number of solutions if the two equations represent the same line.
- The slope-intercept form of a linear equation (y = mx + c) is significant because it allows us to easily compare the slopes and y-intercepts of two linear equations.