Problems 1-2 : Determine whether the following systems of linear equations have one solution.
Problem 1 :
y = 2x + 5
y = 3x - 2
Problem 2 :
y = -2x + 5
y = -2x + 3
Problems 3-4 : Determine whether the following systems of linear equations have no solution.
Problem 3 :
y = 3x + 5
y = 3x - 2
Problem 4 :
4x + 2y - 1 = 0
2x + y + 5 = 0
Problems 5-7 : Determine whether the following systems of linear equations have infinitely many solutions.
Problem 5 :
y = 3x + 5
y = 3x + 5
Problem 6 :
4x + 2y - 1 = 0
2x + y - 0.5 = 0
Problem 7 :
2x - y = 1
4x + y = 5
Problem 8 :
In the following system of linear equations, k is a constant and x and y are variables. For what value of k will the system of equations have unique solution?
kx - y = 4
10x - 5y = 7
Problem 9 :
In the following system of linear equations, k is a constant and x and y are variables. For what value of k will the system of equations have no solution?
kx - 3y = 4
4x - 5y = 7
Problem 10 :
In the following system of linear equations, k is a constant and x and y are variables. For what value of k will the system of equations have infinitely many solution?
kx - 3y = 12
4x - 5y = 20
1. Answer :
y = 2x + 5
y = 3x - 2
y = 2x + 5 ----> slope m = 2
y = 3x - 2 ----> slope m = 3
In the above two linear equations, the slopes are different.
So, the lines intersect in only one point.
Hence, the system has unique solution.
2. Answer :
y = -2x + 5
y = -2x + 3
y = -2x + 5 ----> slope m = -2
y = -2x + 1 ----> slope m = -2
In the above two linear equations, slopes are equal.
Hence, the system has no unique solution.
3. Answer :
y = 3x + 5
y = 3x - 2
y = 3x + 5 ----> m = 3 and b = 5
y = 3x - 2 ----> m = 3 and b = -2
In the above two linear equations, the slope is same and y-intercepts are different.
So, the lines are parallel and they never intersect.
Hence, the system has no solution.
4. Answer :
4x + 2y - 1 = 0
2x + y + 5 = 0
The equations are not in slope-intercept form.
Write them in slope-intercept form.
4x + 2y - 1 = 0 2y = -4x + 1 y = -2x + 1/2 |
2x + y + 5 = 0 y = -2x - 5 |
y = -2x + 1/2 ----> m = -2 and b = 1/2
y = -2x - 5 ----> m = -2 and b = -5
In the given two linear equations, the slope is same and y-intercepts are different.
So, the lines are parallel and they never intersect.
Hence, the system has no solution.
5. Answer :
y = 3x + 5
y = 3x + 5
y = 3x + 5 ----> m = 3 and b = 5
y = 3x + 5 ----> m = 3 and b = 5
In the above two linear equations, both the slopes and y-intercepts are same.
So, the lines coincide and they touch each other in all the points on the line.
Hence, the system has infinitely many solution.
6. Answer :
4x + 2y - 1 = 0
2x + y - 0.5 = 0
The equations are not in slope-intercept form.
Write them in slope-intercept form.
4x + 2y - 1 = 0 2y = -4x + 1 y = -2x + 1/2 |
2x + y - 0.5 = 0 y = -2x + 0.5 y = -2x + 1/2 |
y = -2x + 1/2 ----> m = -2 and b = 1/2
y = -2x + 1/2 ----> m = -2 and b = 1/2
In the given two linear equations, both the slopes and y-intercepts are same.
So, the lines coincide and they touch each other in all the points on the line.
Hence, the system has infinitely many solution.
7. Answer :
2x - y = 1
4x + y = 5
The equations are not in slope-intercept form.
Write them in slope-intercept form.
2x - y = 1 -y = -2x + 1 y = 2x - 1 ---> m = 2 |
4x + y = 5 y = -4x + 5 ----> m = -4 |
In the given two linear equations, the slope are different.
So, the lines do not coincide.
Hence, the system does not have infinitely many solutions.
8. Answer :
kx - y = 4
10x - 5y = 7
The equations are not in slope-intercept form.
Write them in slope-intercept form.
kx - y = 4 -y = -kx + 4 y = kx - 4 |
10x - 5y = 7 -5y = -10x + 7 5y = 10x - 7 y = 2x - 7/5 |
y = kx - 4/3 ----> slope m = k
y = 2x - 7/5 ----> slope m = 2
If the system has unique solution, then the slopes must not be equal.
k ≠ 2
When k ≠ 2, the system has unique solution.
9. Answer :
kx - 3y = 4
4x - 5y = 7
The equations are not in slope-intercept form.
Write them in slope-intercept form.
kx - 3y = 4 -3y = -kx + 4 3y = kx - 4 y = (k/3)x - 4/3 |
4x - 5y = 7 -5y = -4x + 7 5y = 4x - 7 y = (4/5)x - 7/5 |
y = (k/3)x - 4/3 ----> m = k/3 and b = -4/3
y = (4/5)x - 7/5 ----> m = 4/5 and b = -7/5
In the given two linear equations, y-intercepts are different.
If slopes are equal, then the lines will be parallel and they will never intersect. And also, the system will not have solution.
It is given that the system has no solution.
So, the slopes must be equal.
k/3 = 4/5
Multiply both sides by 3.
k = 12/5
When k = 12/5, the system will have no solution.
10. Answer :
kx - 3y = 12
4x - 5y = 20
The equations are not in slope-intercept form.
Write them in slope-intercept form.
kx - 3y = 12 -3y = -kx + 12 3y = kx - 12 y = (k/3)x - 4 |
4x - 5y = 20 -5y = -4x + 20 5y = 4x - 20 y = (4/5)x - 4 |
y = (k/3)x - 4 ----> m = k/3 and b = -4
y = (4/5)x - 4 ----> m = 4/5 and b = -4
In the given two linear equations, y-intercepts are equal.
If slopes also are equal, then the lines will coincide and the system will have infinitely many solutions.
It is given that the system has infinitely many solutions.
So, the slopes must be equal.
k/3 = 4/5
Multiply both sides by 3.
k = 12/5
When k = 12/5, the system will have infinitely many solutions.
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