SYSTEM OF LINEAR EQUATIONS WITH NO SOLUTION WORKSHEET

Problems 1-5 : Determine whether the following systems of linear equations have no solution.

Problem 1 :

y = 3x + 5

y = 3x - 2

Problem 2 :

y = -2x + 1

y = -2x + 1

Problem 3 :

y = 3x - 2 

3y = 9x

Problem 4 :

4x + 2y - 1 = 0

2x + y + 5 = 0

Problem 5 :

2x - y = 1

4x + y = 5

Problem 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 no solution?

kx - 3y = 4

4x - 5y = 7

1. Solution :

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.

2. Solution :

y = -2x + 1

y = -2x + 1

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

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

In the above two linear equations, both the slopes and 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.

3. Solution :

y = 3x - 2 

3y = 9x

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 slope is same and y-intercepts are different.

So, the lines are parallel and they never intersect.

Hence, the system has no solution.

4. Solution :

4x + 2y - 1 = 0

2x + y + 5 = 0

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 - 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. Solution :

2x - y = 1

4x + y = 5

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 only one solution.

6. Solution :

kx - 3y = 4

4x - 5y = 7

The equations are not in slope-intercept form.

Write them in slope-intercept form.

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.

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