Algebra - Linear equations (1 and 2 variables) and inequalities

1

A line has slope -1/2 and another line has slope 2. Are the lines parallel, perpendicular, or neither?

Medium

ExplanationTwo lines are parallel if their slopes are equal, and perpendicular if their slopes multiply to −1. Here, -1/2 and 2 are perpendicular.

2

Solve the system for y: -x + 9y = 174, x − 8y = -155

Medium

ExplanationSolving the system by elimination or substitution gives y = 19 (and x = -3).

3

Solve for x: 6x − 12 ≥ 120

Medium

Explanation6x ≥ 120 − (-12) = 132, so x ≥ 22.

4

Solve the system for y: x − 4y = -20, -5x + 2y = -8

Medium

ExplanationSolving the system by elimination or substitution gives y = 6 (and x = 4).

5

What is the slope of the line through (7, 2) and (4, -6)?

Medium

ExplanationSlope = (y2 − y1)/(x2 − x1) = (-6 − 2)/(4 − 7) = -8/-3 = 8/3.

6

A delivery service charges a fixed fee of Rs. 48 plus Rs. 3 per item. If a customer has a budget of Rs. 96, what is the maximum number of items they can order?

Medium

Explanation48 + 3n ≤ 96, so n ≤ 48/3 ≈ 16.00, meaning at most 16 items.

7

How many solutions does the equation 4x + -15 = 4x + -10 have?

Medium

ExplanationThe x-terms cancel leaving a false statement (since -15 ≠ -10), so there is no solution.

8

What is the slope of the line through (-12, -1) and (-3, 11)?

Medium

ExplanationSlope = (y2 − y1)/(x2 − x1) = (11 − -1)/(-3 − -12) = 12/9 = 4/3.

9

What is the slope of the line through (-6, -12) and (-5, -1)?

Medium

ExplanationSlope = (y2 − y1)/(x2 − x1) = (-1 − -12)/(-5 − -6) = 11/1 = 11.

10

Solve for x: x + 28 = 34

Medium

Explanationx = 34 − (28) = 6, so x = 6/1 = 6.

11

What is the slope of the line through (-8, -1) and (-9, 3)?

Medium

ExplanationSlope = (y2 − y1)/(x2 − x1) = (3 − -1)/(-9 − -8) = 4/-1 = -4.

12

Solve for x: 55 < 9x − 8 < 127

Medium

ExplanationSubtracting -8 from all parts: 63 < 9x < 135. Dividing by 9: 7 < x < 15.

13

What is the y-intercept of the line 12x + 14y = -196?

Medium

ExplanationSet x = 0: 14y = -196, so y = -196/14 = -14.

14

Solve for x: 7x + 29 = 12x + 64

Medium

ExplanationMoving variable terms to one side: (7−12)x = 64−(29), so x = (35)/(-5) = -7.

15

What is the y-intercept of the line 11x + 14y = 56?

Medium

ExplanationSet x = 0: 14y = 56, so y = 56/14 = 4.

16

A delivery service charges a fixed fee of Rs. 50 plus Rs. 28 per item. If a customer has a budget of Rs. 144, what is the maximum number of items they can order?

Medium

Explanation50 + 28n ≤ 144, so n ≤ 94/28 ≈ 3.36, meaning at most 3 items.

17

Solve for x: -x + 26 > 31

Medium

ExplanationDividing both sides by the negative coefficient -1 reverses the inequality sign, giving x < -5.

18

A cinema sells adult tickets for Rs. 5 and child tickets for Rs. 3. If 78 tickets were sold for a total of Rs. 290, how many adult tickets were sold?

Medium

ExplanationLet a = adult tickets, c = child tickets. a + c = 78 and 5a + 3c = 290. Solving gives a = 28, c = 50.

19

Solve for x: x − 7 ≤ -31

Medium

Explanationx ≤ -31 − (-7) = -24, so x ≤ -24.

20

Solve for x: -10x − 27 = -3x − 13

Medium

ExplanationMoving variable terms to one side: (-10−-3)x = -13−(-27), so x = (14)/(-7) = -2.

21

A line has slope 5/6 and another line has slope 5/6. Are the lines parallel, perpendicular, or neither?

Medium

ExplanationTwo lines are parallel if their slopes are equal, and perpendicular if their slopes multiply to −1. Here, 5/6 and 5/6 are parallel.

22

Solve for x: 12x + 11 ≤ -193

Medium

Explanation12x ≤ -193 − (11) = -204, so x ≤ -17.

23

Solve the system for y: 4x − 4y = -16, -2x − 3y = -67

Medium

ExplanationSolving the system by elimination or substitution gives y = 15 (and x = 11).

24

Solve for x: -7x + 30 < 37

Medium

ExplanationDividing both sides by the negative coefficient -7 reverses the inequality sign, giving x > -1.

25

If 2x + y = 43 and 9x + y = 148, what is x + y?

Medium

ExplanationSolving the system gives x = 15 and y = 13, so x + y = 28.

26

Solve for x: -13x − 5 < -109

Medium

ExplanationDividing both sides by the negative coefficient -13 reverses the inequality sign, giving x > 8.

27

Solve for x: -2x − 40 = -88

Medium

Explanation-2x = -88 − (-40) = -48, so x = -48/-2 = 24.

28

A line has slope -7/6 and another line has slope 6/7. Are the lines parallel, perpendicular, or neither?

Medium

ExplanationTwo lines are parallel if their slopes are equal, and perpendicular if their slopes multiply to −1. Here, -7/6 and 6/7 are perpendicular.

29

What is the x-intercept of the line 10x + 3y = 10?

Medium

ExplanationSet y = 0: 10x = 10, so x = 10/10 = 1.

30

A delivery service charges a fixed fee of Rs. 36 plus Rs. 8 per item. If a customer has a budget of Rs. 82, what is the maximum number of items they can order?

Medium

Explanation36 + 8n ≤ 82, so n ≤ 46/8 ≈ 5.75, meaning at most 5 items.

31

Solve the system for y: -2x + 7y = 44, 9x − 6y = 57

Medium

ExplanationSolving the system by elimination or substitution gives y = 10 (and x = 13).

32

Solve for x: -x + 24 < 1

Medium

ExplanationDividing both sides by the negative coefficient -1 reverses the inequality sign, giving x > 23.

33

Solve for x: -6(x + 10) = 6

Medium

ExplanationDividing both sides by -6: x + 10 = -1, so x = -11.

34

If -8x + 4y = -20 and 2x + 9y = -145, what is x + y?

Medium

ExplanationSolving the system gives x = -5 and y = -15, so x + y = -20.

35

A jar contains only Rs. 5 and Rs. 10 coins. There are 47 coins with a total value of Rs. 455. How many Rs. 10 coins are there?

Medium

ExplanationLet x = Rs.5 coins, y = Rs.10 coins. x + y = 47 and 5x + 10y = 455. Solving gives x = 3, y = 44.

36

Solve for x: x/-4 + 19 = 30

Medium

Explanationx/-4 = 30 − (19) = 11, so x = -4 × 11 = -44.

37

Solve for x: -7x + 6 = -4x − 24

Medium

ExplanationMoving variable terms to one side: (-7−-4)x = -24−(6), so x = (-30)/(-3) = 10.

38

Solve for x: 9(x + 0) = 225

Medium

ExplanationDividing both sides by 9: x + 0 = 25, so x = 25.

39

What is the value of -x − 12 when x = -5?

Medium

Explanation-1(-5) − 12 = 5 − 12 = -7.

40

A line has slope 4/7 and passes through (49, -7). What is its y-intercept?

Medium

ExplanationUsing y = mx + b: -7 = (4/7)(49) + b, so b = -7 − 28 = -35.

41

A jar contains only Rs. 5 and Rs. 10 coins. There are 60 coins with a total value of Rs. 375. How many Rs. 10 coins are there?

Medium

ExplanationLet x = Rs.5 coins, y = Rs.10 coins. x + y = 60 and 5x + 10y = 375. Solving gives x = 45, y = 15.

42

Solve for x: -9x + 37 = 55

Medium

Explanation-9x = 55 − (37) = 18, so x = 18/-9 = -2.

43

Solve for x: -7x − 22 = -6x − 10

Medium

ExplanationMoving variable terms to one side: (-7−-6)x = -10−(-22), so x = (12)/(-1) = -12.

44

Solve for x: 5x − 15 ≤ -75

Medium

Explanation5x ≤ -75 − (-15) = -60, so x ≤ -12.

45

Solve for x: x/8 + x/3 = 55

Medium

ExplanationCombining terms: x(1/8 + 1/3) = 55, so x × (11)/24 = 55, giving x = 120.

46

Solve for x: 4(x + 1) = 48

Medium

ExplanationDividing both sides by 4: x + 1 = 12, so x = 11.

47

What is the value of 9x − 7 when x = -16?

Medium

Explanation9(-16) − 7 = -144 − 7 = -151.

48

A line has slope 5/6 and passes through (-48, -3). What is its y-intercept?

Medium

ExplanationUsing y = mx + b: -3 = (5/6)(-48) + b, so b = -3 − -40 = 37.

49

What is the slope of the line through (16, 9) and (26, 15)?

Medium

ExplanationSlope = (y2 − y1)/(x2 − x1) = (15 − 9)/(26 − 16) = 6/10 = 3/5.

50

Solve for x: x/9 + x/3 = 72

Medium

ExplanationCombining terms: x(1/9 + 1/3) = 72, so x × (12)/27 = 72, giving x = 162.

51

A jar contains only Rs. 5 and Rs. 10 coins. There are 37 coins with a total value of Rs. 200. How many Rs. 10 coins are there?

Medium

ExplanationLet x = Rs.5 coins, y = Rs.10 coins. x + y = 37 and 5x + 10y = 200. Solving gives x = 34, y = 3.

52

Solve for x: 116 < 10x + 6 < 196

Medium

ExplanationSubtracting 6 from all parts: 110 < 10x < 190. Dividing by 10: 11 < x < 19.

53

Solve for x: -8(x + 4) = 48

Medium

ExplanationDividing both sides by -8: x + 4 = -6, so x = -10.

54

If -9x + 3y = -147 and 5x − 6y = 112, what is x + y?

Medium

ExplanationSolving the system gives x = 14 and y = -7, so x + y = 7.

55

How many solutions does the equation 10x + -17 = 10x + -17 have?

Medium

ExplanationBoth sides are identical for every x, so every value of x works — infinitely many solutions.

56

How many solutions does the equation 10x + -2 = 6x + -16 have?

Medium

ExplanationSince the coefficients of x differ, the equation has exactly one solution.

57

A cinema sells adult tickets for Rs. 5 and child tickets for Rs. 2. If 85 tickets were sold for a total of Rs. 299, how many adult tickets were sold?

Medium

ExplanationLet a = adult tickets, c = child tickets. a + c = 85 and 5a + 2c = 299. Solving gives a = 43, c = 42.

58

Solve for x: -4x + 1 > -47

Medium

ExplanationDividing both sides by the negative coefficient -4 reverses the inequality sign, giving x < 12.

59

What is the y-intercept of the line 13x + 7y = -84?

Medium

ExplanationSet x = 0: 7y = -84, so y = -84/7 = -12.

60

Solve for x: 4(x + 14) = 152

Medium

ExplanationDividing both sides by 4: x + 14 = 38, so x = 24.

61

Solve for x: 10x − 4 > -204

Medium

Explanation10x > -204 − (-4) = -200, so x > -20.

62

Solve for x: -6(x + 4) = -8(x + 6)

Medium

ExplanationExpanding both sides and collecting x terms gives x = -12.

63

Solve for x: -59 < 3x − 5 < -26

Medium

ExplanationSubtracting -5 from all parts: -54 < 3x < -21. Dividing by 3: -18 < x < -7.

64

Solve for x: 5(x + 17) = 5

Medium

ExplanationDividing both sides by 5: x + 17 = 1, so x = -16.

65

Given 7x + 10y = 10, what is y when x = -40?

Medium

Explanation10y = 10 − 7(-40) = 10 − -280 = 290, so y = 290/10 = 29.

66

Solve for x: -49 < 3x − 7 < -31

Medium

ExplanationSubtracting -7 from all parts: -42 < 3x < -24. Dividing by 3: -14 < x < -8.

67

Solve for x: 2(x − 17) = -82

Medium

ExplanationDividing both sides by 2: x − 17 = -41, so x = -24.

68

Solve for x: 4x + 15 = 99

Medium

Explanation4x = 99 − (15) = 84, so x = 84/4 = 21.

69

What is the slope of the line through (3, 3) and (-5, -11)?

Medium

ExplanationSlope = (y2 − y1)/(x2 − x1) = (-11 − 3)/(-5 − 3) = -14/-8 = 7/4.

70

Solve for x: 44 < 2x + 14 < 70

Medium

ExplanationSubtracting 14 from all parts: 30 < 2x < 56. Dividing by 2: 15 < x < 28.

71

Solve the system for x: 8x + 5y = 51, 9x − 7y = 272

Medium

ExplanationSolving the system by elimination or substitution gives x = 17 (and y = -17).

72

Solve the system for x: -6x + 4y = -148, 2x + 6y = -2

Medium

ExplanationSolving the system by elimination or substitution gives x = 20 (and y = -7).

73

Solve for x: -2x + 29 = 3

Medium

Explanation-2x = 3 − (29) = -26, so x = -26/-2 = 13.

74

How many solutions does the equation 4x + 11 = 5x + 12 have?

Medium

ExplanationSince the coefficients of x differ, the equation has exactly one solution.

75

Solve for x: -6x + 5 = -85

Medium

Explanation-6x = -85 − (5) = -90, so x = -90/-6 = 15.

76

Solve for x: x/8 + x/6 = -42

Medium

ExplanationCombining terms: x(1/8 + 1/6) = -42, so x × (14)/48 = -42, giving x = -144.

77

Solve for x: 9x − 23 > -158

Medium

Explanation9x > -158 − (-23) = -135, so x > -15.

78

Solve for x: -10x − 16 = 274

Medium

Explanation-10x = 274 − (-16) = 290, so x = 290/-10 = -29.

79

A jar contains only Rs. 5 and Rs. 10 coins. There are 54 coins with a total value of Rs. 490. How many Rs. 10 coins are there?

Medium

ExplanationLet x = Rs.5 coins, y = Rs.10 coins. x + y = 54 and 5x + 10y = 490. Solving gives x = 10, y = 44.

80

A delivery service charges a fixed fee of Rs. 37 plus Rs. 4 per item. If a customer has a budget of Rs. 139, what is the maximum number of items they can order?

Medium

Explanation37 + 4n ≤ 139, so n ≤ 102/4 ≈ 25.50, meaning at most 25 items.

81

A cinema sells adult tickets for Rs. 23 and child tickets for Rs. 6. If 42 tickets were sold for a total of Rs. 592, how many adult tickets were sold?

Medium

ExplanationLet a = adult tickets, c = child tickets. a + c = 42 and 23a + 6c = 592. Solving gives a = 20, c = 22.

82

Solve for x: 1 < 5x + 16 < 61

Medium

ExplanationSubtracting 16 from all parts: -15 < 5x < 45. Dividing by 5: -3 < x < 9.

83

Solve for x: -7x − 11 < -53

Medium

ExplanationDividing both sides by the negative coefficient -7 reverses the inequality sign, giving x > 6.

84

What is the value of -9x + 4 when x = 11?

Medium

Explanation-9(11) + 4 = -99 + 4 = -95.

85

A delivery service charges a fixed fee of Rs. 41 plus Rs. 30 per item. If a customer has a budget of Rs. 213, what is the maximum number of items they can order?

Medium

Explanation41 + 30n ≤ 213, so n ≤ 172/30 ≈ 5.73, meaning at most 5 items.

86

What is the x-intercept of the line 7x + 11y = -77?

Medium

ExplanationSet y = 0: 7x = -77, so x = -77/7 = -11.

87

Solve for x: -x + 29 = -2x + 17

Medium

ExplanationMoving variable terms to one side: (-1−-2)x = 17−(29), so x = (-12)/(1) = -12.

88

What is the value of 15x + 8 when x = -7?

Medium

Explanation15(-7) + 8 = -105 + 8 = -97.

89

What is the x-intercept of the line 3x + 11y = -18?

Medium

ExplanationSet y = 0: 3x = -18, so x = -18/3 = -6.

90

Solve the system for y: -5x − 5y = -5, 7x + 3y = -17

Medium

ExplanationSolving the system by elimination or substitution gives y = 6 (and x = -5).

91

What is the slope of the line through (-19, -18) and (-13, 1)?

Medium

ExplanationSlope = (y2 − y1)/(x2 − x1) = (1 − -18)/(-13 − -19) = 19/6 = 19/6.

92

Solve for x: x + 32 = 39

Medium

Explanationx = 39 − (32) = 7, so x = 7/1 = 7.

93

Solve the system for x: -x − 7y = -25, 6x − 7y = -46

Medium

ExplanationSolving the system by elimination or substitution gives x = -3 (and y = 4).

94

Solve for x: -52 < 4x + 12 < -20

Medium

ExplanationSubtracting 12 from all parts: -64 < 4x < -32. Dividing by 4: -16 < x < -8.

95

Solve for x: -5x + 36 = 11

Medium

Explanation-5x = 11 − (36) = -25, so x = -25/-5 = 5.

96

What is the value of -12x + 3 when x = -6?

Medium

Explanation-12(-6) + 3 = 72 + 3 = 75.

97

Solve the system for y: 3x + 5y = -111, -2x − 7y = 118

Medium

ExplanationSolving the system by elimination or substitution gives y = -12 (and x = -17).

98

Solve the system for x: -x + y = -23, 5x + 4y = 79

Medium

ExplanationSolving the system by elimination or substitution gives x = 19 (and y = -4).

99

What is the value of 11x − 3 when x = -13?

Medium

Explanation11(-13) − 3 = -143 − 3 = -146.

100

Solve for x: 10x + 26 < 186

Medium

Explanation10x < 186 − (26) = 160, so x < 16.