Pair of Linear Equations in Two Variables · Class 10 Mathematics: 80 flashcards
All 80 flashcards from this kit, free and with no sign-up.
1. What is the general form of a pair of linear equations in two variables?
Show the answer
$a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$.
2. What is a solution of a pair of linear equations?
Show the answer
A pair of values (x, y) that satisfies both equations.
3. How many points do you need to draw a line?
Show the answer
Two.
4. How do you find the two easiest points of $x + 3y = 6$?
Show the answer
$x = 0$ gives $(0, 2)$; $y = 0$ gives $(6, 0)$.
5. Where is the solution on the graph?
Show the answer
At the point where the two lines cross.
6. What does it mean if the two lines coincide?
Show the answer
Infinitely many solutions — the equations are equivalent.
7. What does it mean if the two lines are parallel?
Show the answer
No solution — the pair is inconsistent.
8. Define a consistent pair.
Show the answer
A pair that has at least one solution.
9. Define an inconsistent pair.
Show the answer
A pair with no solution — parallel lines.
10. Is a dependent pair consistent?
Show the answer
Yes — it has infinitely many solutions.
11. 2 pencils and 3 erasers cost ₹9. Write the equation.
Show the answer
$2x + 3y = 9$, with ₹x per pencil and ₹y per eraser.
12. Write 'the number of skirts is two less than twice the pants' as an equation.
Show the answer
$y = 2x - 2$, with x pants and y skirts.
13. Solve graphically: $x + 3y = 6$ and $2x - 3y = 12$.
Show the answer
The lines cross at $(6, 0)$, so $x = 6$, $y = 0$.
14. How do you check a solution?
Show the answer
Substitute it into both original equations and see that both hold.
15. Why is a graph a poor tool for $7x - 15y = 2$, $x + 2y = 3$?
Show the answer
The solution is $\left(\frac{49}{29}, \frac{19}{29}\right)$ — impossible to read accurately.
16. Can two different straight lines cross at two points?
Show the answer
No. If they share two points they are the same line.
17. What is the first line of a word-problem answer?
Show the answer
'Let x be …' and 'Let y be …'
18. A pair of equations has a solution. What is the pair called?
Show the answer
Consistent.
19. How many solutions does one linear equation in two variables have?
Show the answer
Infinitely many — every point on its line.
20. The lines $y = 2x - 2$ and $y = 4x - 4$ cross at which point?
Show the answer
(1, 0)
21. Before using the ratio test, what must you do?
Show the answer
Write both equations in the form $ax + by + c = 0$, carrying the signs.
22. Condition for exactly one solution?
Show the answer
$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$
23. Condition for infinitely many solutions?
Show the answer
$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
24. Condition for no solution?
Show the answer
$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$
25. Which picture goes with one solution?
Show the answer
Intersecting lines.
26. Which picture goes with infinitely many solutions?
Show the answer
Coincident lines — the same line twice.
27. Which picture goes with no solution?
Show the answer
Parallel lines.
28. In $2x + 3y = 9$, what is c in standard form?
Show the answer
−9, since the equation is $2x + 3y - 9 = 0$.
29. Classify $x - 2y = 0$ and $3x + 4y - 20 = 0$.
Show the answer
$\frac13 \neq \frac{-2}{4}$, so one solution — consistent.
30. Classify $2x + 3y - 9 = 0$ and $4x + 6y - 18 = 0$.
Show the answer
All three ratios are ½ — infinitely many solutions, dependent.
31. Classify $x + 2y - 4 = 0$ and $2x + 4y - 12 = 0$.
Show the answer
½ = 2/4 but −4/−12 = ⅓ — no solution, inconsistent.
32. What does 'consistent' mean?
Show the answer
The pair has at least one solution.
33. What does 'dependent' mean?
Show the answer
The equations are equivalent; infinitely many solutions, and always consistent.
34. Can a pair be dependent and inconsistent?
Show the answer
No. Dependent pairs always have solutions.
35. Why check the third ratio when finding k?
Show the answer
Equal a and b ratios alone leave both coincident and parallel possible.
36. What replaces the ratio test when a coefficient is zero?
Show the answer
Cross-products: $a_1b_2 = a_2b_1$.
37. Find k so that $2x + 3y - 7 = 0$ and $(k-1)x + (k+2)y - 3k = 0$ have infinitely many solutions.
Show the answer
k = 7, and the c ratio checks out.
38. For a unique solution, do the constants matter?
Show the answer
No — only $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ is needed.
39. In matrix terms, what does $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ say?
Show the answer
That $a_1b_2 - a_2b_1 \neq 0$.
40. State the three names for the three cases in order.
Show the answer
Consistent (one), dependent-consistent (infinite), inconsistent (none).
41. What are the three steps of the substitution method?
Show the answer
Make one variable the subject; substitute it into the other equation and solve; substitute back for the second variable.
42. Which equation do you substitute into?
Show the answer
The OTHER one — not the equation you rearranged.
43. Which variable should you make the subject?
Show the answer
One with coefficient 1 or −1, to avoid fractions.
44. Rearrange $x + 2y = 3$ for x.
Show the answer
$x = 3 - 2y$
45. Substituting $x = 3 - 2y$ into $7x - 15y = 2$ gives what first line?
Show the answer
$7(3 - 2y) - 15y = 2$
46. Solve $7x - 15y = 2$, $x + 2y = 3$.
Show the answer
$x = \frac{49}{29}$, $y = \frac{19}{29}$
47. After substituting, you get $18 = 18$. What does it mean?
Show the answer
Infinitely many solutions — the equations describe the same line.
48. After substituting, you get $18 = 20$. What does it mean?
Show the answer
No solution — the lines are parallel.
49. What is $7(3 - 2y)$?
Show the answer
$21 - 14y$
50. A two-digit number with tens digit x and units digit y is written how?
Show the answer
$10x + y$; reversed it is $10y + x$.
51. 'The number is 27 more than its reverse' becomes what simplified equation?
Show the answer
$x - y = 3$, from $9x - 9y = 27$.
52. Aftab's ages problem gives which pair?
Show the answer
$x - 7y = -42$ and $x - 3y = 6$.
53. Solve $x - 7y = -42$ and $x - 3y = 6$.
Show the answer
x = 42, y = 12.
54. Where should you verify your answer?
Show the answer
In both ORIGINAL equations.
55. Is a fractional answer a sign of a mistake?
Show the answer
No — many pairs have fractional solutions.
56. If a question says 'solve by substitution', can you use elimination?
Show the answer
No — the method marks are for the named method.
57. Why not substitute back into the same equation?
Show the answer
It gives an identity like 3 = 3, which carries no new information.
58. After finding y, how do you get x fastest?
Show the answer
Put y into the Step 1 expression you already wrote.
59. Solve $x + y = 14$, $x - y = 4$ by substitution.
Show the answer
$x = 14 - y$, so $14 - 2y = 4$, y = 5 and x = 9.
60. Which method does a coefficient of 1 favour?
Show the answer
Substitution.
61. What are the three steps of elimination?
Show the answer
Match one variable's coefficients; add or subtract to cancel it; solve and substitute back.
62. When do you subtract the equations?
Show the answer
When the matched coefficients have the SAME sign.
63. When do you add them?
Show the answer
When the matched coefficients have opposite signs.
64. What is $3 \times (2x + 3y = 13)$?
Show the answer
$6x + 9y = 39$ — the constant is multiplied too.
65. How do you choose the multipliers?
Show the answer
From the LCM of the two coefficients of the variable you are removing.
66. Solve $2x + 3y = 13$ and $5x - 2y = 4$.
Show the answer
x = 2, y = 3.
67. Which variable should you eliminate?
Show the answer
The one whose coefficients have the smaller LCM.
68. Elimination ends with $0 = 0$. What does it mean?
Show the answer
Infinitely many solutions — the lines coincide.
69. Elimination ends with $0 = 2$. What does it mean?
Show the answer
No solution — the lines are parallel.
70. Write a two-digit number with tens digit x and units digit y.
Show the answer
$10x + y$; reversed, $10y + x$.
71. 'Five years ago' changes an age x to what?
Show the answer
$x - 5$ — and every age in the problem changes too.
72. A boat of speed u in a stream of speed v goes downstream at what speed?
Show the answer
$u + v$; upstream it is $u - v$.
73. NCERT's 9 : 7 income problem gives which incomes?
Show the answer
₹18,000 and ₹14,000.
74. Why may you multiply an equation by a constant?
Show the answer
A non-zero multiple is an equivalent equation with the same solutions.
75. What are the three marked parts of a word problem?
Show the answer
Defining the letters, forming and solving the equations, answering in words.
76. Solve $x + y = 5$ and $2x - 3y = 4$.
Show the answer
x = 3.8, y = 1.2 — exactly, $x = \frac{19}{5}$ and $y = \frac{6}{5}$.
77. Which method suits $7x + 9y = 11$ and $4x - 5y = 3$?
Show the answer
Elimination — no coefficient is 1, so substitution would bring fractions.
78. Where do you substitute back?
Show the answer
Into whichever ORIGINAL equation has the simpler numbers.
79. What is the difference between a two-digit number and its reverse always a multiple of?
Show the answer
9, because $(10x + y) - (10y + x) = 9(x - y)$.
80. How do you finish a word problem?
Show the answer
With a sentence naming the quantity and its units.
Your notes can be a kit like these
Every kit here was made from real study material. Point Veda at a chapter, a PDF, a photo or a link and get notes, flashcards, practice MCQs, games and a scored test — free, in 42 languages.
- Chapter notes
- Flashcards
- Practice MCQs
- Scored tests
- Exam prep
Get Veda free See what students get
Free to start · No ads · Works in 42 languages