Maths ch3 linear equations

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Quick notes

A pair of linear equations: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.

Graphical method: the lines may intersect (one solution), be parallel (no solution), or coincide (infinitely many solutions).

a₁/a₂ ≠ b₁/b₂ → intersecting, unique solution, consistent.
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → parallel, no solution, inconsistent.
a₁/a₂ = b₁/b₂ = c₁/c₂ → coincident, infinitely many solutions, dependent (consistent).

Substitution: write one variable in terms of the other and substitute. Elimination: make the coefficients of one variable equal, then add or subtract to remove it.

Part A: Multiple choice 16 questions

1. A pair of linear equations has a unique solution when the lines

are parallel
coincide
intersect at one point
are both vertical

Intersecting lines meet at exactly one point.

2. If the two lines are parallel, the pair of equations has

no solution
one solution
two solutions
infinitely many solutions

Parallel lines never meet.

3. If the two lines coincide, the pair has

no solution
exactly one solution
exactly two solutions
infinitely many solutions

Every point on the line is a common solution.

4. If a₁/a₂ ≠ b₁/b₂, the pair of equations is

inconsistent
consistent with a unique solution
dependent
parallel

Unequal ratios of x and y coefficients mean the lines intersect.

5. The solution of x + y = 5 and x − y = 1 is

x = 2, y = 3
x = 4, y = 1
x = 3, y = 2
x = 1, y = 4

Adding: 2x = 6, so x = 3 and y = 2.

6. The pair 2x + 3y = 9 and 4x + 6y = 18 has

infinitely many solutions
no solution
a unique solution
exactly two solutions

2/4 = 3/6 = 9/18 = 1/2, so the lines coincide.

7. The solution of x − 2y = 0 and 3x + 4y = 20 is

x = 2, y = 4
x = 4, y = 2
x = 6, y = 3
x = 8, y = −1

x = 2y, so 6y + 4y = 20, y = 2 and x = 4.

8. For which value of k does kx + 3y = k − 3, 12x + ky = k have no solution?

6
3
−3
−6

k/12 = 3/k gives k = ±6. For k = 6 all ratios equal 1/2 (infinitely many). For k = −6 the third ratio is 3/2, so no solution.

9. The graph of x = 3 is a line

parallel to the x-axis
parallel to the y-axis
through the origin
along the x-axis

x stays 3 for every y, so it is a vertical line.

10. The pair 3x + 2y = 5 and 2x − 3y = 7 is

consistent with a unique solution
inconsistent
dependent
parallel

3/2 ≠ 2/(−3), so the lines intersect.

11. A single linear equation in two variables has

no solution
exactly one solution
exactly two solutions
infinitely many solutions

Every point on its line is a solution.

12. If x = 2, y = 3 is a solution of 2x − 3y + a = 0, then a is

−5
5
13
−13

4 − 9 + a = 0 gives a = 5.

13. The sum of two numbers is 20 and their difference is 4. The numbers are

14 and 6
11 and 9
12 and 8
16 and 4

x + y = 20, x − y = 4 gives x = 12, y = 8.

14. 10 students took part in a quiz. The number of girls is 4 more than the number of boys. How many girls?

g + b = 10 and g − b = 4, so g = 7.

15. Subtracting 2x − y = 1 from 2x + y = 7 gives

x = 2
y = 2
y = 3
x = 3

(2x + y) − (2x − y) = 7 − 1, so 2y = 6 and y = 3.

16. The lines 2x + 3y = 5 and 4x + 6y = 7 are

intersecting
coincident
perpendicular
parallel

2/4 = 3/6 = 1/2 but 5/7 ≠ 1/2.

Part B: Fill in the blanks 8 blanks

17. If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are [[parallel]].

18. A pair of linear equations that has no solution is called [[inconsistent]].

19. Solving x + y = 14 and x − y = 4 gives x = [[9]] and y = [[5]].

20. In the substitution method, we express one variable in terms of the [[other]] variable.

21. The graph of y = 0 is the [[x-axis|x axis|xaxis]].

22. 5 pencils and 7 pens cost ₹50, and 7 pencils and 5 pens cost ₹46. One pencil costs ₹[[3]] and one pen costs ₹[[5]].

Part C: Questions and answers 9 questions

23. Solve by substitution: s − t = 3 and s/3 + t/2 = 6.

From the first equation, s = t + 3.
(t + 3)/3 + t/2 = 6 → 2(t + 3) + 3t = 36 → 5t = 30 → t = 6.
So s = 9. s = 9, t = 6.

24. Solve by elimination: 3x + 4y = 10 and 2x − 2y = 2.

Multiply the second equation by 2: 4x − 4y = 4.
Add to the first: 7x = 14, so x = 2.
Then 2(2) − 2y = 2 gives y = 1. x = 2, y = 1.

25. Is the pair 5x − 4y + 8 = 0 and 7x + 6y − 9 = 0 consistent?

a₁/a₂ = 5/7 and b₁/b₂ = −4/6 = −2/3.
Since 5/7 ≠ −2/3, the lines intersect. The pair is consistent with a unique solution.

26. A fraction becomes 9/11 if 2 is added to both numerator and denominator. If 3 is added to both, it becomes 5/6. Find the fraction.

Let the fraction be x/y.
(x + 2)/(y + 2) = 9/11 → 11x − 9y = −4.
(x + 3)/(y + 3) = 5/6 → 6x − 5y = −3.
Multiply: 55x − 45y = −20 and 54x − 45y = −27. Subtract: x = 7, then y = 9.
Fraction = 7/9.

27. Five years hence, Jacob's age will be three times his son's age. Five years ago, Jacob's age was seven times his son's age. Find their present ages.

Let Jacob = x, son = y.
x + 5 = 3(y + 5) → x − 3y = 10.
x − 5 = 7(y − 5) → x − 7y = −30.
Subtract: 4y = 40, so y = 10 and x = 40.
Jacob is 40 years, son is 10 years.

28. Half the perimeter of a rectangular garden is 36 m, and its length is 4 m more than its width. Find the dimensions.

l + w = 36 and l − w = 4.
Adding: 2l = 40, so l = 20 and w = 16.
Length 20 m, width 16 m.

29. A taxi charges a fixed amount plus a charge per km. For 10 km the fare is ₹105, and for 15 km it is ₹155. What is the fare for 25 km?

Let fixed charge = f and rate = k per km.
f + 10k = 105 and f + 15k = 155. Subtract: 5k = 50, so k = 10 and f = 5.
Fare for 25 km = 5 + 25 × 10 = ₹255.

30. For what value of k does 2x + 3y = 7 and (k − 1)x + (k + 2)y = 3k have infinitely many solutions?

We need 2/(k − 1) = 3/(k + 2) = 7/(3k).
2(k + 2) = 3(k − 1) gives k = 7.
Check: 2/6 = 3/9 = 7/21 = 1/3 ✓. So k = 7.

31. What does it mean, in terms of graphs, if a pair of linear equations has no solution?

The two lines are parallel. They have the same slope but different intercepts, so they never meet and there is no common point.