SOLVING SYSTEMS BY ELIMINATION WORKSHEET

Problem 1 :

Solve by elimination method. 

3x + 4y  =  -25

2x - 3y  =  6

Problem 2 :

Solve by elimination method

2x + 3y  =  5

3x + 4y  =  7

Problem 3 :

Solve the system of equations using elimination method. Check the solution by graphing.

2x - 3y  =  12

x + 3y  =  6

Problem 4 : 

A park charges $10 for adults and $5 for kids. How many adults tickets and kids tickets were sold, if a total of 548 tickets were sold for a total of $3750 ? 

Problem 5 :

Sum of the cost price of two products is $50. Sum of the selling price of the same two products is $52. If one is sold at 20% profit and other one is sold at 20% loss, find the cost price of each product. 

Detailed Answer Key

Problem 1 :

Solve by elimination method. 

3x + 4y  =  -25

2x - 3y  =  6

Solution :

3x + 4y  =  -25 ---- (1)

2x - 3y  =  6  ---- (2)

Both x terms and y terms have different coefficients in the above system of equations.

Let's try to make the coefficients of y terms equal.

To make the coefficients of y terms equal, we have to find the least common multiple 4 and 3.

The least common multiple of 4 and 3 is 12.

Multiply the first equation by 3 in order to make the coefficient of y as 12 and multiply the second equation by 4 in order to make the coefficient of y as -12. 

(1) ⋅ 3 ---->  9x + 12y  =  -75

(2) ⋅ 4 ---->  8x - 12y  =  24

Now, we can add the two equations and eliminate y as shown below. 

Divide each side by 17. 

x  =  -3

Substitute -3 for x in (1). 

(1)---->  3(-3) + 4y  =  -25

-9 + 4y  =  -25

Add 9 to each side.

4y  =  -16

Divide each side by 4.

y  =  -4

So, the values of x and y are -3 and -4 respectively.

Problem 2 :

Solve by elimination method

2x + 3y  =  5

3x + 4y  =  7

Solution :

2x + 3y  =  5  ----(1)

3x + 4y  =  7  ----(2)

Both x terms and y terms have different coefficients in the above system of equations.

Let's try to make the coefficients of x terms equal.

To make the coefficients of y terms equal, we have to find the least common multiple 2 and 3.

The least common multiple of 2 and 3 is 6.

Multiply the first equation by 3 in order to make the coefficient of x as 6 and multiply the second equation by -2 in order to make the coefficient of x as -6. 

(1) ⋅ 3 ---->  6x + 9y  =  15

(2) ⋅ -2 ----> -6x - 8y  =  -14

Now, we can add the two equations and eliminate x as shown below. 

Substitute 1 for y in (1). 

(1)---->  2x + 3(1)  =  5

2x + 3  =  5

Subtract 3 from each side. 

2x  =  2

Divide each side by 2.

x  =  1

So, the values of x and y are 1 and 1 respectively.

Problem 3 :

Solve the system of equations using elimination method. Check the solution by graphing.

2x - 3y  =  12

x + 3y  =  6

Solution : 

2x - 3y  =  12 -----(1)

x + 3y  =  6 -----(2)

Step 1 :

In the given two equations, the variable y is having the same coefficient. And also, the variable y is having different signs. 

So we can eliminate the variable y by adding the two equations.  

Divide both sides by 3. 

3x / 3  =  18 / 3

x  =  6

Step 2 : 

Substitute 6 for x in (2).

(2)-----> 6 + 3y  =  6 

Subtract 6 from each side.

3y  =  0

Divide each side by 3.

3y / 3  =  0

y  =  0

Step 3 : 

Write the solution as ordered pair. 

(x, y)  =  (6, 0)

Step 4 : 

Check the solution by graphing. 

To graph the equations, write them in slope-intercept form.

That is, 

The point of intersection is (6, 0).

Problem 4 : 

A park charges $10 for adults and $5 for kids. How many adults tickets and kids tickets were sold, if a total of 548 tickets were sold for a total of $3750 ? 

Solution : 

Step 1 :

Let "x" be the number of adults tickets and "y" be the number of kids tickets.

No. of adults tickets + No. of kids tickets  =  Total

x + y  =  548 ----(1)

Step 2 : 

Write an equation which represents the total cost.

Cost of "x" no. adults tickets  =  10x

Cost of "y" no. of kids tickets  =  5y

Total cost  =  $3750

Then, we have 

10x + 5y  =  3750

 Divide both sides by 5.

2x + y  =  750 ----(2)

Step 3 :

Solve (1) and (2) using elimination method. 

x + y  =  548 ----(1)

2x + y  =  750 ----(2)

In the above two equations, y is having the same coefficient, that is 1. 

Multiply the first equation by -1 to get the coefficient of -1. And keep the second equation as it is. 

Then, we have 

-x - y  =  - 548

2x + y  =  750

We can add the above two equations and eliminate y. 

Then, we have

x  =  202

Step 4 : 

Substitute 202 for x in the first equation.

(1)---->  202 + y  =  548

Subtract 202 from each side.

y  =  306

So, the number of adults tickets sold is 202 and the number of kids tickets sold is 346.

Problem 5 :

Sum of the cost price of two products is $50. Sum of the selling price of the same two products is $52. If one is sold at 20% profit and other one is sold at 20% loss, find the cost price of each product.  

Solution :

Step 1 :

Let 'x' and 'y' be the cost prices of two products. 

Then,  

x + y  =  50 -----(1)

Step 2 :

Let us assume that 'x' is sold at 20% profit.

Then, the selling price of 'x' is

=  120% of 'x'

=  1.2x

Let us assume that 'y' is sold at 20% loss.

Then, the selling price of 'y' is

=  80% of 'y'

=  0.8y

Given : Selling price of 'x' + Selling price of 'y' = 52.

Then, 

1.2x + 0.8y  =  52

To get rid decimal, multiply both sides by 10.

12x + 8y  =  520

Divide both sides by 4.

3x + 2y  =  130 -----(2)

Step 3 : 

Eliminate one of the variables to get the value of the other variable.

In (1) and (2), both the variables 'x' and 'y' do not have the same coefficient.

One of the variables must have the same coefficient. 

So, multiply both sides of (1) by 2 to make the coefficients of 'y' same with different signs in the equations.  

(1) ⋅ 2 ----->  2x + 2y  =  100 -----(3) 

Variable 'y' has the same sign in both (2) and (3). 

To change the sign of 'y' in (3), multiply both sides of (3) by negative sign.

- (2x + 2y)  =  - 100

- 2x - 2y  =  - 100 -----(4)

Step 4 : 

Now, eliminate the variable 'y' in (2) and (4) as shown below and find the value of x'. 

Step 5 : 

Substitute 30 for x in (1) to get the value of y. 

(1)-----> 30 + y  =  50

Subtract 30 from both sides. 

y  =  20

So, the cost prices of two products are $30 and $20.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.