SYSTEM OF EQUATIONS WITH INFINITELY MANY SOLUTIONS

Examples 1-5 : Determine whether the following systems of linear equations have infinitely many solutions.

Example 1 :

y = 3x + 5

y = 3x + 5

Solution :

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.

Example 2 :

y = -2x - 5

y = -2x + 1

Solution :

y = -2x - 5 ----> m = -2 and b = -5

y = -2x + 1 ----> m = -2 and b = 1

In the above two linear equations, the slopes are equal, but y-intercepts are different.

So, the lines do not coincide.

Hence, the system does not have infinitely many solution.

Example 3 :

y = 3x - 2 

3y = 9x

Solution :

y = 3x - 2 ----> m = 3 and b = -2

The second equation 3y = 9x is not in slope intercept form. Divide both sides by 3 to get the equation in slope- intercept form.

y = 3x

y = 3x ----> m = 3 and b = 0

In the given two linear equations, the slopes are equal, but y-intercepts are different.

So, the lines do not coincide.

Hence, the system does not have infinitely many solution.

Example 4 :

4x + 2y - 1 = 0

2x + y - 0.5 = 0

Solution :

The equations are not in slope-intercept form.

Write them in slope-intercept form.

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.

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 do not coincide.

Hence, the system does not have infinitely many solutions.

Example 6 :

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

Solution :

The equations are not in slope-intercept form.

Write them in slope-intercept form.

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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