DETERMINING NUMBER OF SOLUTIONS WORKSHEET

Use the properties of equality to simplify the equation given below. Say whether the equation has one, zero, or infinitely many solutions.

Problem 1 :

4x - 3  =  2x + 13

Solution

Problem 2 :

4x - 5  =  2(2x - 1) - 3

Solution

Problem 3 :

4x + 2  =  4x - 5

Solution

Problem 4 :

The equation P = 2.5m + 35 represents the price P (in dollars) of a bracelet, where m is the cost of the materials (in dollars). The price of a bracelet is $115. What is the cost of the materials?

Solution

Problem 5 :

A 455-foot fence encloses a pasture. What is the length of each side of the pasture?

Solution

Problem 6 :

A machine prints 230 movie posters each hour. Write and solve an equation to find the number of hours it takes the machine to print 1265 posters.

Solution

Problem 7 :

Use the table to write and solve an equation to find the number of points p you need to score in the fourth game so that the mean number of points is 20?

Solution

Problem 8 :

A boat travels x miles per hour upstream on the Mississippi River. On the return trip, the boat travels 2 miles per hour faster. How far does the boat travel upstream?

Solution

Problem 9 :

One-sixth of the girls and two-sevenths of the boys in a school marching band are in the percussion section. The percussion section has 6 girls and 10 boys. How many students are in the marching band? Explain.

Solution

Problem 1 :

Which of the following pairs of linear equations has unique solution, no solution, infinitely many solutions. In case there is unique solution, find it by using cross multiplication method.

(i)   x – 3y – 3  =  0

     3x – 9y – 2  =  0

(ii)  2x + y  =  5

      3x + 2y  =  8

(iii) 3x – 5y  =  20

       6x – 10y  =  40

(iv) x – 3y – 7  =  0

     3x – 3y – 15  =  0

Solution

Problem 2 :

The pair of linear equations 2𝑥 = 5y + 6 and 15y = 6𝑥 - 8 represents two lines which are

(a)  Intersecting    (b) Parallel   (c) coincident

(d) either Intersecting or Parallel

Solution

Problem 3 :

If the pair of linear equations 𝑥 - y = 1, 𝑥 + ky = 5 has a unique solution 𝑥 = 2, y = 1 then the value of k is

(a) -2    (b) -3    (c) 3     (d) 4

Solution

Problem 4 :

The pair of linear equations 3𝑥 + 5y = 3 and 6𝑥 + ky = 8 do not have a solution if k

(a) = 5   (b) = 10   (c) ≠10    (d) ≠ 5

Solution

Determine whether the following systems of linear equations have unique solution.

Problem 1 :

y = 2x + 5

y = 3x - 2

Solution

Problem 2 :

y = -2x + 5

y = -2x + 3

Solution

Determine whether the following systems of linear equations have no solution.

Problem 3 :

y = 3x + 5

y = 3x - 2

Solution

Problem  4 :

4x + 2y - 1 = 0

2x + y + 5 = 0

Solution

Determine whether the following systems of linear equations have infinitely many solutions.

Problem 5 :

y = 3x + 5

y = 3x + 5

Solution

Problem 6 :

4x + 2y - 1 = 0

2x + y - 0.5 = 0

Solution

Problem 7 :

2x - y = 1

4x + y = 5

Solution

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

Solution

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

Solution

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

Solution

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