Problems 1-5 : 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
Problem 3 :
y = 3x - 2
3y = 6x
Problem 4 :
3x + y - 1 = 0
2x + y + 5 = 0
Problem 5 :
2x - y = 1
4x + y = 5
Problem 6 :
Determine whether the following system of equations has one solution. If so, find the solution.
y = 3x - 1
y = 2x - 5
Problem 7 :
Determine whether the following system of equations has one solution. If so, find the solution.
y = -2x + 9
y = 5x + 9
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 one solution?
kx - y = 4
10x - 5y = 7
1. Solution :
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 one solution.
2. Solution :
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 or one solution.
3. Solution :
y = 3x - 2
3y = 6x
y = 3x - 2 ----> slope m = 3
The second equation 3y = 6x is not in slope intercept form. Divide both sides by 3 to get the equation in slope- intercept form.
y = 2x
y = 2x ----> slope m = 2
In the given two linear equations, the slopes are different.
So, the lines intersect in only one point.
Hence, the system has one solution.
4. Solution :
3x + y - 1 = 0
2x + y + 5 = 0
The equations are not in slope-intercept form.
Write them in slope-intercept form.
3x + y - 1 = 0 y = -3x + 1 |
2x + y + 5 = 0 y = -2x - 5 |
y = -3x + 1 ----> slope m = -3
y = -2x - 5 ----> slope m = -2
In the given two linear equations, the slopes are different.
So, the lines intersect in only one point.
Hence, the system has one solution.
5. Solution :
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 |
4x + y = 5 y = -4x + 5 |
In the given two linear equations, the slope are different.
So, the lines intersect in only one point.
Hence, the system has one solution.
6. Solution :
y = 3x - 1
y = 2x - 5
y = 3x - 1 ----> slope m = 3
y = 2x - 5 ----> slope m = 2
In the given two linear equations, the slopes are different.
So, the lines intersect in only one point.
Hence, the system has one solution.
Solving for x and y :
y = 3x - 1 ----(1)
y = 2x - 5 ----(2)
Substitute '3x - 1' for y in (2).
3x - 1 = 2x - 5
x = -4
Substitute x = -4 in (1).
y = 3(-4) - 1
y = -12 - 1
y = -13
Therefore, the solution is
x = -4 and y = -13
7. Solution :
y = -2x + 9
y = 5x + 9
y = -2x + 9 ----> slope m = -2
y = 5x + 9 ----> slope m = 5
In the given two linear equations, the slopes are different.
So, the lines intersect in only one point.
Hence, the system has one solution.
There is same y-intercept '9' in both the equations.
So, the point of intersection of the two lines is y-intercept, that is (0, 9)
Therefore, the solution is
x = 0 and y = 9
8. Solution :
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 one solution.
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