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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