How Many Solutions Does This Linear System Have?1. $y = 2x - 5$2. − 8 X − 4 Y = − 20 -8x - 4y = -20 − 8 X − 4 Y = − 20 A. One Solution: ( − 2.5 , 0 (-2.5, 0 ( − 2.5 , 0 ] B. One Solution: ( 2.5 , 0 (2.5, 0 ( 2.5 , 0 ] C. No Solution D. Infinite Number Of Solutions
Introduction
Linear systems are a fundamental concept in mathematics, and understanding how to solve them is crucial for various applications in science, engineering, and economics. A linear system consists of two or more linear equations that are solved simultaneously to find the values of the variables. In this article, we will explore how to determine the number of solutions for a linear system, using the given equations as a case study.
Understanding the Equations
We are given two linear equations:
- y = 2x - 5
- -8x - 4y = -20
To determine the number of solutions, we need to first understand the nature of these equations. The first equation is in the slope-intercept form, where the slope (m) is 2 and the y-intercept (b) is -5. This equation represents a straight line with a slope of 2 and a y-intercept of -5.
The second equation is in the standard form, where the coefficients of x and y are -8 and -4, respectively. This equation can be rewritten as:
-8x - 4y = -20
To make it easier to work with, we can multiply both sides of the equation by -1, which gives us:
8x + 4y = 20
Graphical Representation
To visualize the number of solutions, we can graph the two equations on a coordinate plane. The first equation, y = 2x - 5, is a straight line with a slope of 2 and a y-intercept of -5. We can plot this line on the coordinate plane by finding two points that satisfy the equation.
The second equation, 8x + 4y = 20, can be rewritten as:
y = -2x + 5
This equation represents a straight line with a slope of -2 and a y-intercept of 5. We can plot this line on the coordinate plane by finding two points that satisfy the equation.
Determining the Number of Solutions
To determine the number of solutions, we need to find the intersection point of the two lines. If the lines intersect at a single point, then the system has one solution. If the lines are parallel and do not intersect, then the system has no solution. If the lines coincide, then the system has an infinite number of solutions.
Let's analyze the two equations:
- y = 2x - 5
- y = -2x + 5
We can see that the two equations have the same slope (2 and -2) and the same y-intercept (-5 and 5). This means that the two lines coincide, and the system has an infinite number of solutions.
Conclusion
In conclusion, the linear system consisting of the two equations y = 2x - 5 and -8x - 4y = -20 has an infinite number of solutions. This is because the two equations represent the same line, and there are an infinite number of points on this line that satisfy both equations.
Solving the System
To solve the system, we can use either of the two equations. Let's use the first equation:
y = 2x - 5
We can substitute this expression for y into the second equation:
-8x - 4(2x - 5) = -20
Expanding and simplifying the equation, we get:
-8x - 8x + 20 = -20
Combine like terms:
-16x + 20 = -20
Subtract 20 from both sides:
-16x = -40
Divide both sides by -16:
x = 2.5
Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:
y = 2x - 5
Substitute x = 2.5:
y = 2(2.5) - 5
Simplify the equation:
y = 5 - 5
y = 0
Therefore, the solution to the system is (2.5, 0).
Final Answer
The final answer is:
- D. Infinite number of solutions
Note: The other options are incorrect because the system has an infinite number of solutions, not one solution, no solution, or a single solution.
Introduction
Linear systems are a fundamental concept in mathematics, and understanding how to solve them is crucial for various applications in science, engineering, and economics. A linear system consists of two or more linear equations that are solved simultaneously to find the values of the variables. In this article, we will explore how to determine the number of solutions for a linear system, using the given equations as a case study.
Understanding the Equations
We are given two linear equations:
- y = 2x - 5
- -8x - 4y = -20
To determine the number of solutions, we need to first understand the nature of these equations. The first equation is in the slope-intercept form, where the slope (m) is 2 and the y-intercept (b) is -5. This equation represents a straight line with a slope of 2 and a y-intercept of -5.
The second equation is in the standard form, where the coefficients of x and y are -8 and -4, respectively. This equation can be rewritten as:
-8x - 4y = -20
To make it easier to work with, we can multiply both sides of the equation by -1, which gives us:
8x + 4y = 20
Graphical Representation
To visualize the number of solutions, we can graph the two equations on a coordinate plane. The first equation, y = 2x - 5, is a straight line with a slope of 2 and a y-intercept of -5. We can plot this line on the coordinate plane by finding two points that satisfy the equation.
The second equation, 8x + 4y = 20, can be rewritten as:
y = -2x + 5
This equation represents a straight line with a slope of -2 and a y-intercept of 5. We can plot this line on the coordinate plane by finding two points that satisfy the equation.
Determining the Number of Solutions
To determine the number of solutions, we need to find the intersection point of the two lines. If the lines intersect at a single point, then the system has one solution. If the lines are parallel and do not intersect, then the system has no solution. If the lines coincide, then the system has an infinite number of solutions.
Let's analyze the two equations:
- y = 2x - 5
- y = -2x + 5
We can see that the two equations have the same slope (2 and -2) and the same y-intercept (-5 and 5). This means that the two lines coincide, and the system has an infinite number of solutions.
Conclusion
In conclusion, the linear system consisting of the two equations y = 2x - 5 and -8x - 4y = -20 has an infinite number of solutions. This is because the two equations represent the same line, and there are an infinite number of points on this line that satisfy both equations.
Solving the System
To solve the system, we can use either of the two equations. Let's use the first equation:
y = 2x - 5
We can substitute this expression for y into the second equation:
-8x - 4(2x - 5) = -20
Expanding and simplifying the equation, we get:
-8x - 8x + 20 = -20
Combine like terms:
-16x + 20 = -20
Subtract 20 from both sides:
-16x = -40
Divide both sides by -16:
x = 2.5
Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:
y = 2x - 5
Substitute x = 2.5:
y = 2(2.5) - 5
Simplify the equation:
y = 5 - 5
y = 0
Therefore, the solution to the system is (2.5, 0).
Final Answer
The final answer is:
- D. Infinite number of solutions
Q&A
Q: What is a linear system?
A: A linear system is a set of two or more linear equations that are solved simultaneously to find the values of the variables.
Q: How do I determine the number of solutions for a linear system?
A: To determine the number of solutions, you need to find the intersection point of the two lines. If the lines intersect at a single point, then the system has one solution. If the lines are parallel and do not intersect, then the system has no solution. If the lines coincide, then the system has an infinite number of solutions.
Q: What is the difference between a linear system and a nonlinear system?
A: A linear system consists of linear equations, while a nonlinear system consists of nonlinear equations. Linear equations have a constant slope, while nonlinear equations have a variable slope.
Q: Can a linear system have an infinite number of solutions?
A: Yes, a linear system can have an infinite number of solutions if the two lines coincide.
Q: How do I solve a linear system?
A: To solve a linear system, you can use either of the two equations. You can substitute one expression for y into the other equation and solve for x. Then, you can substitute the value of x into one of the original equations to find the value of y.
Q: What is the final answer for the given linear system?
A: The final answer is:
- D. Infinite number of solutions
Q: Can you provide an example of a linear system with one solution?
A: Yes, here is an example of a linear system with one solution:
- y = 2x - 3
- -4x - 2y = -6
To solve this system, you can use either of the two equations. Let's use the first equation:
y = 2x - 3
We can substitute this expression for y into the second equation:
-4x - 2(2x - 3) = -6
Expanding and simplifying the equation, we get:
-4x - 4x + 6 = -6
Combine like terms:
-8x + 6 = -6
Subtract 6 from both sides:
-8x = -12
Divide both sides by -8:
x = 1.5
Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:
y = 2x - 3
Substitute x = 1.5:
y = 2(1.5) - 3
Simplify the equation:
y = 3 - 3
y = 0
Therefore, the solution to the system is (1.5, 0).
Q: Can you provide an example of a linear system with no solution?
A: Yes, here is an example of a linear system with no solution:
- y = 2x - 3
- -4x - 2y = -10
To solve this system, you can use either of the two equations. Let's use the first equation:
y = 2x - 3
We can substitute this expression for y into the second equation:
-4x - 2(2x - 3) = -10
Expanding and simplifying the equation, we get:
-4x - 4x + 6 = -10
Combine like terms:
-8x + 6 = -10
Subtract 6 from both sides:
-8x = -16
Divide both sides by -8:
x = 2
Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:
y = 2x - 3
Substitute x = 2:
y = 2(2) - 3
Simplify the equation:
y = 4 - 3
y = 1
However, if we substitute x = 2 into the second equation, we get:
-4(2) - 2(1) = -10
Simplify the equation:
-8 - 2 = -10
-10 = -10
This is a true statement, which means that the system has no solution.
Q: Can you provide an example of a linear system with an infinite number of solutions?
A: Yes, here is an example of a linear system with an infinite number of solutions:
- y = 2x - 5
- -8x - 4y = -20
To solve this system, you can use either of the two equations. Let's use the first equation:
y = 2x - 5
We can substitute this expression for y into the second equation:
-8x - 4(2x - 5) = -20
Expanding and simplifying the equation, we get:
-8x - 8x + 20 = -20
Combine like terms:
-16x + 20 = -20
Subtract