SYSTEM OF LINEAR EQUATIONS WORKSHEET

Questions 1-3 : Solve the linear systems by substitution method.

Question 1 :

y = x - 1

2x + y = 5

Question 2 :

-2x + y = 4

x - y = -1

Question 3 :

x - 2y = 1

3x + 5y = 14

Questions 4-6 : Solve the linear systems by elimination method.

Question 4 :

2x - 3y = 1

5x + 3y = 13

Question 5 :

5x + 6y = 17

3x - 4y = -5

Question 6 :

x - 5y = -2

3x + 2y = 11

Question 7 :

The age of a father is three more than two times the age of his son. If the sum of ages of the father and son is equal to 78, find their ages.

Question 8 :

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

Questions 9-11 : Check whether the following systems of linear equations have one solution, infinitely many solutions or no solution.

Question 9 :

3x - y + 4 = 0

5x + y + 2 = 0

Question 10 :

2x + y - 3 = 0

4x + 2y = 6

Question 11 :

3x + y = 4

3x + y - 5 = 0

1. Answer :

y = x - 1 ----(1)

2x + y = 5 ----(2)

Substitute y = x - 1 in (2).

2x + (x - 1) = 5

2x + x - 1 = 5

3x - 1 = 5

Add 1 to both sides.

3x = 6

Divide both sides by 3.

x = 2

Substitute x = 2 in (1).

y = 2 - 1

= 1

The solution is (x, y) = (2, 1).

2. Answer :

-2x + y = 4

x - y = -1

Solve for y in -2x + y = 4.

-2x + y = 4

Add 2x to both sides.

y = 4 + 2x ----(1)

x - y = -1 ----(2)

Substitute y = 4 + 2x in (2).

x - (4 + 2x) = -1

x - 4 - 2x = -1

-x - 4 = -1

Add 4 to both sides.

-x = 3

Multiply both sides by -1.

x = -3

Substitute x = -3 in (1).

y = 2(-3) + 4

= -6 + 4

= -2

The solution is (x, y) = (-3, -2).

3. Answer :

x - 2y = 1

3x + 5y = 14

Solve for x in x - 2y = 1.

x - 2y = 1

Add 2y to both sides.

x = 1 + 2y ----(1)

3x + 5y = 14 ----(2)

Substitute x = 1 + 2y in (2).

3(1 + 2y) + 5y = 14.

3 + 6y + 5y = 14

3 + 11y = 14

Subtract 3 from both sides.

11y = 11

Divide both sides by 11.

y = 1

Substitute y = 1 in (1).

x = 1 + 2(1)

= 1 + 2

= 3

The solution is (x, y) = (3, 1).

4. Answer :

2x - 3y = 1

5x + 3y = 13

In the given system of linear equations, we have the same coefficient for y with different signs. By adding the two equations, y can be eliminated.

2x - 3y = 1 ----(1)

5x + 3y = 13 ----(2)

(1) + (2) :

7x = 14

Divide both sides by 7.

x = 2

Substitute x = 2Divide both sides by 3.

x = 2

Substitute x = 2 in (1).

2(2) - 3y = 1

4 - 3y = 1

Subtract 4 from both sides.

-3y = -3

Divide both sides by -3.

y = 1

The solution is (x, y) = (2, 1).

5. Answer :

5x + 6y = 17 ----(1)

3x - 4y = -5 ----(2)

In the given system of linear equations, both the variables x and y do not have the same coefficient. If we want to eliminate y, it must have the same coefficient with different signs. We can make the coefficient of y same in both the equations using least common multiple.

Least common multiple of (6, 4) = 12.

(1) x 2 + (2) x 3 :

19x = 19

Divide both sides by 19.

x = 1

Substitute x = 1 in (1).

5(1) + 6y = 17

5 + 6y = 17

Subtract 5 from both sides.

6y = 12

Divide both sides by 6.

y = 2

The solution is (x, y) = (1, 2).

6. Answer :

x - 5y = -2 ----(1)

3x + 2y = 11 ----(2)

In the given system of linear equations, both the variables x and y do not have the same coefficient. If we want to eliminate x, it must have the same coefficient with different signs. We make the coefficient of x same in both the equations by multiplying the first equation by 3.

(1) x 3 - (2) ;

-17y = -17

Divide both sides by -17.

y = 1

Substitute y = 1 in (1).

x - 5(1) = -2

x - 5 = -2

Add 5 to both sides.

x = 3

The solution is (x, y) = (3, 1).

7. Answer :

Let f and s be the ages of father and son respectively.

f = 2s + 3 ----(1)

f + s = 78 ----(2)

Substitute f = 2s + 3 in (2).

2s + 3 + s = 78

3s + 3 = 78

Subtract 3 from both sides.

3s = 75

Divide both sides by 3.

s = 25

Substitute s = 25 in (1).

f = 2(25) + 3

f = 50 + 3

f = 53

The age of the father is 53 years and that of the son is 25 years.

8. Answer :

Let x and y be the number of adult kids tickets sold respectively.

Given :A total of 548 tickets were sold.

x + y = 548 ----(1)

Given : Cost of each adult ticket is $10 and kid ticket is $5 and tickets were sold for a total of $3750.

10x + 5y = 3750

Divide both sides by 5.

2x + y = 750 ----(2)

In (1) and (2), we have the same coefficient for y. By subtracting the equation, we can eliminate y.

(2) - (1) :

x = 202

Substitute x = 202 in (1).

202 + y = 548

Subtract 202 from both sides.

y = 346

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

9. Answer :

3x - y + 4 = 0

5x + y + 2 = 0

Write the given two linear equations in slope intercept form, that is 

y = mx + b

where m is the slope b is the y-intercept.

3x - y + 4 = 0

-y = -3x - 4

y = 3x + 4

5x + y - 2 = 0

y = -5x + 2  

y = 3x + 4 ----> m = 3 and b = 4

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

The given two lines have different slopes (y-intercepts may be same or different), then the system has one solution.

10. Answer :

2x + y - 3 = 0

4x + 2y = 6

Write the given two linear equations in slope intercept form.

2x + y - 3 = 0

y = -2x + 3

4x + 2y = 6

2y = -4x + 6

y = -2x + 3

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

The given two lines have the same slope and same y-intercept, then the system has infinitely many solutions.

11. Answer :

3x + y = 4

3x + y - 5 = 0

Write the given two linear equations in slope intercept form.

3x + y = 4

y = -3x + 4

3x + y - 5 = 0

y = -3x + 5  

y = -3x + 4 ----> m = -3 and b = 4

y = -3x + 5 ----> m = -2 and b = 5

The given two lines have same slope and different y-intercepts, then the lines are parallel and they will never intersect. So, the  system has no solution.

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