SYSTEM OF LINEAR EQUATIONS WITH UNIQUE SOLUTION

Examples 1-5 : Determine whether the following systems of linear equations have unique solution.

Example 1 :

y = 2x + 5

y = 3x - 2

Solution :

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.

Example 2 :

y = -2x + 5

y = -2x + 3

Solution :

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.

Example 3 :

y = 3x - 2

3y = 6x

Solution :

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 unique solution.

Example 4 :

3x + y - 1 = 0

2x + y + 5 = 0

Solution :

The equations are not in slope-intercept form.

Write them in slope-intercept form.

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 unique solution.

Example 5 :

2x - y = 1

4x + y = 5

Solution :

The equations are not in slope-intercept form.

Write them in slope-intercept form.

In the given two linear equations, the slope are different.

So, the lines intersect in only one point.

Hence, the system has unique solution.

Example 6 :

Determine whether the following system of equations has unique solution. If so, find the solution.

y = 3x - 1

y = 2x - 5

Solution :

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 unique 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

Example 7 :

Determine whether the following system of equations has unique solution. If so, find the solution.

y = -2x + 9

y = 5x + 9

Solution :

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 unique 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

Example 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

Solution :

The equations are not in slope-intercept form.

Write them in slope-intercept form.

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.

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