Create systems of equations in two variables that have (a) no solution, (b) one distinct solution, and (c) infinite solutions. Show that each system has the indicated number of solutions by using a graph or solving the system.
You can do this. Just recall that
(a) parallel lines -- same slope
(b) intersecting lines -- different slopes
(c) the same line -- just write it in two different ways
Certainly! Here are examples of systems of equations that have no solution, one distinct solution, and infinite solutions.
(a) No Solution:
Consider the following system of equations:
Equation 1: 3x - 2y = 5
Equation 2: 6x - 4y = 10
To show that this system has no solution, we can graph these equations. If the lines representing the equations are parallel and don't intersect, there is no solution.
Here is the graph:
The lines are parallel and do not intersect, indicating that there is no solution to this system of equations.
(b) One Distinct Solution:
Consider the following system of equations:
Equation 1: 2x + y = 5
Equation 2: 3x - y = 7
To show that this system has one distinct solution, we can solve it by either graphing or using an algebraic method like substitution or elimination.
Using the substitution method:
From Equation 1, solve for y:
y = 5 - 2x
Substitute this value into Equation 2:
3x - (5 - 2x) = 7
3x - 5 + 2x = 7
5x - 5 = 7
5x = 12
x = 12/5
Now substitute x back into Equation 1:
2(12/5) + y = 5
24/5 + y = 5
y = 5 - 24/5
y = 1/5
So the solution to this system is x = 12/5 and y = 1/5.
(c) Infinite Solutions:
Consider the following system of equations:
Equation 1: 2x + 3y = 6
Equation 2: 4x + 6y = 12
To show that this system has infinitely many solutions, we can solve it by either graphing or algebraic methods.
Using the algebraic method of elimination:
Multiply Equation 1 by 2 and Equation 2 by -1 to eliminate x:
4x + 6y = 12
-4x - 6y = -12
When we add these equations together, the x terms cancel out:
0 = 0
This equation indicates that the statement is always true, regardless of the values of x and y. That means that any values of x and y that satisfy one equation will also satisfy the other equation. Thus, this system has infinitely many solutions.
I hope this helps to demonstrate systems of equations with no solution, one distinct solution, and infinite solutions!
To create systems of equations in two variables that have (a) no solution, (b) one distinct solution, and (c) infinite solutions, we will walk through each scenario and explain how to create the equations and determine the solutions.
(a) No Solution:
To create a system of equations with no solution, we want two lines that are parallel and will never intersect. The general form of equations for two lines is: y = mx + b, where m represents the slope and b represents the y-intercept.
Let's consider the following system of equations:
Equation 1: y = 2x + 3
Equation 2: y = 2x + 5
Both equations have the same slope (2), but different y-intercepts (3 and 5). When we graph these equations, we will see that the lines never intersect, indicating that there is no solution.
(b) One Distinct Solution:
In this scenario, we want a system of equations where the two lines intersect at a single point. This occurs when the lines have different slopes.
For instance, consider the following system of equations:
Equation 1: y = 3x + 2
Equation 2: y = -2x + 5
The slopes of the two equations are different (3 and -2). Graphing these equations will show that they intersect at a single point, representing a distinct solution.
(c) Infinite Solutions:
To create a system of equations with infinite solutions, the two equations must be equivalent, meaning they represent the same line.
Consider the following system of equations:
Equation 1: y = -4x + 2
Equation 2: 2y = -8x + 4
We can see that Equation 2 is just Equation 1 multiplied by 2 on both sides. This means that both equations represent the same line. When we graph these equations, the lines coincide perfectly, indicating that there are infinite solutions.
To visually verify these solutions on a graph, plot the equations on the x-y plane. The unique intersection point represents one solution, parallel lines with no intersection represent no solution, and overlapping lines represent infinite solutions.