Basic Algebraic Equations Competitive Exams
100 questions · Mathematics
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Question 1 of 100
Solve for x: 8 - x = 2x + 2
Explanation8 - 2 = 2x + x → 6 = 3x → x = 2.
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Question 2 of 100
Solve: x² = 25
ExplanationTaking the square root of both sides: x = ±5.
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Question 3 of 100
If x + y = 10 and x - y = 2, find the value of x.
ExplanationAdding both equations: 2x = 12, so x = 6.
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Question 4 of 100
Solve for x: 2(x + 3) = 3(x - 1)
Explanation2x + 6 = 3x - 3 → 6 + 3 = 3x - 2x → x = 9.
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Question 5 of 100
Solve for x: 5x + 3 = 2x + 12
Explanation5x - 2x = 12 - 3 → 3x = 9 → x = 3.
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Question 6 of 100
Solve: x² + 2x - 8 = 0
ExplanationFactoring: (x+4)(x-2) = 0, so x = -4 or x = 2.
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Question 7 of 100
Solve: x² + 3x - 4 = 0
ExplanationFactoring: (x+4)(x-1) = 0, so x = -4 or x = 1.
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Question 8 of 100
Determine the nature of the roots of x² + x + 1 = 0.
ExplanationDiscriminant = 1 - 4 = -3, which is negative, so there are no real roots.
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Question 9 of 100
Solve for x: (x/3) + 4 = 9
Explanationx/3 = 9 - 4 = 5, so x = 5 × 3 = 15.
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Question 10 of 100
Using the previous scenario (sum 20, difference 4), find the smaller number.
ExplanationLarger number = 12, so smaller number = 20 - 12 = 8.
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Question 11 of 100
Solve: x² - 5x + 6 = 0
ExplanationFactoring: (x-2)(x-3) = 0, so x = 2 or x = 3.
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Question 12 of 100
If x + y = 15 and x - y = 5, find the value of x.
ExplanationAdding: 2x = 20, so x = 10 (then y = 5).
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Question 13 of 100
A number when added to its half gives 45. Find the number.
ExplanationLet the number be x. x + x/2 = 45 → 3x/2 = 45 → x = 30.
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Question 14 of 100
Solve x² - 3x - 4 = 0 using the quadratic formula.
ExplanationDiscriminant = 9 + 16 = 25, √25 = 5. x = (3 ± 5)/2, giving x = 4 or x = -1.
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Question 15 of 100
If 5x + y = 17 and x + y = 5, find the value of y.
ExplanationSubtracting: 4x = 12 → x = 3, so y = 5 - 3 = 2.
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Question 16 of 100
Solve for x: (3x + 2)/5 = 4
Explanation3x + 2 = 20 → 3x = 18 → x = 6.
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Question 17 of 100
If 4x - 3y = 6 and 2x + y = 8, find the value of x.
Explanationy = 8 - 2x. Substituting: 4x - 3(8-2x) = 6 → 10x = 30 → x = 3 (then y = 2).
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Question 18 of 100
Solve for x: x/4 - 3 = 2
Explanationx/4 = 2 + 3 = 5, so x = 5 × 4 = 20.
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Question 19 of 100
The difference between a number and its one-third is 14. Find the number.
ExplanationLet the number be x. x - x/3 = 14 → 2x/3 = 14 → x = 21.
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Question 20 of 100
Solve: x² - 16 = 0
Explanationx² = 16, so x = ±4.
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Question 21 of 100
Three times a number minus 7 equals twice the number plus 5. Find the number.
ExplanationLet the number be x. 3x - 7 = 2x + 5 → x = 12.
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Question 22 of 100
Find the sum of the roots of 2x² - 5x + 3 = 0.
ExplanationSum of roots = -b/a = -(-5)/2 = 5/2.
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Question 23 of 100
Expand: (a - b)²
ExplanationThe standard identity: (a-b)² = a² - 2ab + b².
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Question 24 of 100
If 5x + 2y = 19 and 3x + 2y = 13, find the value of y.
ExplanationSubtracting: 2x = 6 → x = 3; then 15 + 2y = 19 → y = 2.
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Question 25 of 100
Solve for x: 15 - 2x = x
Explanation15 = x + 2x = 3x, so x = 15 / 3 = 5.
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Question 26 of 100
If x + 3y = 11 and 2x - y = 1, find the value of y.
ExplanationFrom the second equation, y = 2x - 1. Substituting: x + 3(2x-1) = 11 → 7x = 14 → x = 2, so y = 3.
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Question 27 of 100
The cost of 3 pens is Rs. 45. Find the cost of 7 pens at the same rate.
ExplanationCost per pen = 45/3 = Rs. 15. Cost of 7 pens = 7 × 15 = Rs. 105.
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Question 28 of 100
Solve: x² - 9 = 0
Explanationx² = 9, so x = 3 or x = -3, i.e., x = ±3.
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Question 29 of 100
The length of a rectangle is twice its width. If the perimeter is 60 cm, find the width.
ExplanationLet width = w, length = 2w. Perimeter = 2(2w + w) = 6w = 60, so w = 10 cm.
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Question 30 of 100
If a - b = 3 and ab = 4, find a² + b².
Explanation(a-b)² = a² - 2ab + b² → 9 = a² + b² - 8 → a² + b² = 17.
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Question 31 of 100
Solve: x² - 10x + 21 = 0
ExplanationFactoring: (x-3)(x-7) = 0, so x = 3 or x = 7.
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Question 32 of 100
If 3x + 2y = 16 and x + y = 6, find the value of y.
Explanationx = 6 - y. Substituting: 3(6-y) + 2y = 16 → 18 - y = 16 → y = 2.
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Question 33 of 100
Solve: x² - 4x + 4 = 0
ExplanationFactoring: (x-2)² = 0, so x = 2 is a repeated (double) root.
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Question 34 of 100
If x + y = 10 and x - y = 2, find the value of y.
ExplanationSubtracting: 2y = 8, so y = 4 (and x = 6).
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Question 35 of 100
A number increased by 8 equals 20. Find the number.
ExplanationLet the number be x. x + 8 = 20, so x = 12.
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Question 36 of 100
Solve for x: x + (x/2) = 9
Explanation(3x/2) = 9 → 3x = 18 → x = 6.
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Question 37 of 100
If 2x + 3y = 13 and 4x - y = 5, find the value of x.
ExplanationFrom the second equation, y = 4x - 5. Substituting: 2x + 3(4x-5) = 13 → 14x = 28 → x = 2.
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Question 38 of 100
Solve for x: 3x - 8 = 2(x - 1)
Explanation3x - 8 = 2x - 2 → 3x - 2x = -2 + 8 → x = 6.
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Question 39 of 100
Solve for x: 9 - 2x = 3
Explanation-2x = 3 - 9 = -6, so x = -6 / -2 = 3.
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Question 40 of 100
Solve for x: 4x/7 = 12
Explanation4x = 12 × 7 = 84, so x = 84 / 4 = 21.
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Question 41 of 100
Expand: (a + b)²
ExplanationThe standard identity: (a+b)² = a² + 2ab + b².
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Question 42 of 100
Solve for x: 4(x - 2) = 20
Explanation4x - 8 = 20 → 4x = 28 → x = 7.
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Question 43 of 100
If x - y = 3 and x + y = 11, find the value of xy.
ExplanationAdding: 2x = 14 → x = 7; then y = 4. So xy = 7 × 4 = 28.
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Question 44 of 100
Solve for x: 5x + 2 = 3x + 18
Explanation5x - 3x = 18 - 2 → 2x = 16 → x = 8.
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Question 45 of 100
Find the smaller of two consecutive even integers whose sum is 38.
ExplanationLet the integers be x and x+2. x + (x+2) = 38 → 2x = 36 → x = 18.
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Question 46 of 100
Simplify: (a + b)² - (a - b)²
Explanation(a+b)² - (a-b)² = (a²+2ab+b²) - (a²-2ab+b²) = 4ab.
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Question 47 of 100
The sum of two numbers is 20 and their difference is 4. Find the larger number.
ExplanationLet the numbers be x and y. x + y = 20, x - y = 4. Adding: 2x = 24, so x = 12.
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Question 48 of 100
Solve for x: 2x + 5 = 15
Explanation2x = 15 - 5 = 10, so x = 10 / 2 = 5.
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Question 49 of 100
A father is 3 times as old as his son. The sum of their ages is 48. Find the son's age.
ExplanationLet the son's age be x, father's age = 3x. x + 3x = 48 → 4x = 48 → x = 12.
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Question 50 of 100
Using the previous scenario (length is twice the width, perimeter 60 cm), find the length.
ExplanationWidth = 10 cm (from 6w = 60), so length = 2 × 10 = 20 cm.
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Question 51 of 100
The sum of three consecutive integers is 63. Find the middle integer.
ExplanationLet the integers be (x-1), x, (x+1). Their sum is 3x = 63, so x = 21.
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Question 52 of 100
Expand: (x + 5)²
Explanation(x+5)² = x² + 2(5)x + 25 = x² + 10x + 25.
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Question 53 of 100
Find the sum of the roots of x² - 7x + 12 = 0.
ExplanationFor ax² + bx + c = 0, sum of roots = -b/a = -(-7)/1 = 7.
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Question 54 of 100
Solve: 2x² - 5x + 2 = 0
ExplanationFactoring: (2x-1)(x-2) = 0, so x = 1/2 or x = 2.
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Question 55 of 100
If x + 2y = 8 and 2x + y = 7, find the value of x.
Explanationx = 8 - 2y. Substituting: 2(8-2y) + y = 7 → 16 - 3y = 7 → y = 3, so x = 2.
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Question 56 of 100
Solve: x² - x - 6 = 0
ExplanationFactoring: (x-3)(x+2) = 0, so x = 3 or x = -2.
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Question 57 of 100
Solve for x: (2x/5) = 8
Explanation2x = 8 × 5 = 40, so x = 40 / 2 = 20.
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Question 58 of 100
Solve for x: 2x/3 + 1 = 5
Explanation2x/3 = 4 → 2x = 12 → x = 6.
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Question 59 of 100
Solve for x: 6x + 1 = 25
Explanation6x = 25 - 1 = 24, so x = 24 / 6 = 4.
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Question 60 of 100
If x + 1/x = 5, find the value of x² + 1/x².
ExplanationSquaring both sides: (x + 1/x)² = x² + 2 + 1/x² = 25, so x² + 1/x² = 25 - 2 = 23.
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Question 61 of 100
Solve: x² + 7x + 10 = 0
ExplanationFactoring: (x+2)(x+5) = 0, so x = -2 or x = -5.
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Question 62 of 100
Expand: (2x + 3)(x - 1)
Explanation(2x+3)(x-1) = 2x² - 2x + 3x - 3 = 2x² + x - 3.
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Question 63 of 100
Simplify: (x + 3)(x - 3)
ExplanationUsing the difference of squares: (x+3)(x-3) = x² - 9.
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Question 64 of 100
Solve: x² - 2x - 3 = 0
ExplanationFactoring: (x-3)(x+1) = 0, so x = 3 or x = -1.
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Question 65 of 100
If 2x - y = 5 and x + y = 7, find the value of x.
ExplanationAdding both equations: 3x = 12, so x = 4 (then y = 3).
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Question 66 of 100
Solve for x: 2x - 3 = x + 8
Explanation2x - x = 8 + 3 → x = 11.
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Question 67 of 100
If 7x - 2y = 3 and 3x + 2y = 17, find the value of y.
ExplanationAdding: 10x = 20 → x = 2; then 6 + 2y = 17 → y = 5.5.
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Question 68 of 100
Find the product of the roots of 2x² - 5x + 3 = 0.
ExplanationProduct of roots = c/a = 3/2.
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Question 69 of 100
If 2x + 3y = 12 and x - y = 1, find the value of x.
ExplanationFrom x - y = 1, x = y + 1. Substituting: 2(y+1) + 3y = 12 → 5y = 10 → y = 2, so x = 3.
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Question 70 of 100
Twice a number decreased by 3 is 11. Find the number.
ExplanationLet the number be x. 2x - 3 = 11 → 2x = 14 → x = 7.
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Question 71 of 100
Solve: x² - 6x + 9 = 0
ExplanationFactoring: (x-3)² = 0, so x = 3 is a repeated (double) root.
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Question 72 of 100
Solve for x: 10 - 3x = 4
Explanation-3x = 4 - 10 = -6, so x = -6 / -3 = 2.
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Question 73 of 100
If 3x + y = 10 and x + y = 4, find the value of y.
ExplanationSubtracting: 2x = 6 → x = 3, so y = 4 - 3 = 1.
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Question 74 of 100
Solve: x² + 5x + 6 = 0
ExplanationFactoring: (x+2)(x+3) = 0, so x = -2 or x = -3.
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Question 75 of 100
The sum of two consecutive integers is 25. Find the smaller integer.
ExplanationLet the integers be x and x+1. x + (x+1) = 25 → 2x + 1 = 25 → x = 12.
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Question 76 of 100
Solve for x: 3x + 4 = 5x - 10
Explanation4 + 10 = 5x - 3x → 14 = 2x → x = 7.
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Question 77 of 100
The sum of a number and 5 is 12. Find the number.
ExplanationLet the number be x. x + 5 = 12, so x = 7.
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Question 78 of 100
If 2x + y = 11 and x + y = 7, find the value of x.
ExplanationSubtracting the second from the first: x = 4 (then y = 3).
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Question 79 of 100
Find the product of the roots of x² - 7x + 12 = 0.
ExplanationFor ax² + bx + c = 0, product of roots = c/a = 12/1 = 12.
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Question 80 of 100
If 5 apples cost Rs. 60, find the cost of 12 apples at the same rate.
ExplanationCost per apple = 60/5 = Rs. 12. Cost of 12 apples = 12 × 12 = Rs. 144.
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Question 81 of 100
If x/2 + y = 5 and x - y = 1, find the value of x.
Explanationx = y + 1. Substituting: (y+1)/2 + y = 5 → y + 1 + 2y = 10 → y = 3, so x = 4.
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Question 82 of 100
Solve for x: 5(x - 4) = 15
Explanationx - 4 = 15/5 = 3, so x = 3 + 4 = 7.
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Question 83 of 100
Solve: x² - 7x + 12 = 0
ExplanationFactoring: (x-3)(x-4) = 0, so x = 3 or x = 4.
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Question 84 of 100
Determine the nature of the roots of x² - 4x + 4 = 0.
ExplanationDiscriminant = b² - 4ac = 16 - 16 = 0, so the roots are real and equal.
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Question 85 of 100
Five years ago, a man was 4 times as old as his son. If the son is now 15, find the man's current age.
ExplanationFive years ago, son's age = 10, so man's age then = 4 × 10 = 40. Man's current age = 40 + 5 = 45.
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Question 86 of 100
Factorize: a² - b²
ExplanationThe difference of squares identity: a² - b² = (a+b)(a-b).
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Question 87 of 100
If 4x + y = 14 and x + y = 5, find the value of y.
ExplanationSubtracting: 3x = 9 → x = 3, so y = 5 - 3 = 2.
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Question 88 of 100
Determine the nature of the roots of x² - 5x + 6 = 0.
ExplanationDiscriminant = 25 - 24 = 1, which is positive, so there are two distinct real roots.
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Question 89 of 100
A number is 5 more than another. Their sum is 21. Find the smaller number.
ExplanationLet the smaller number be x. x + (x+5) = 21 → 2x = 16 → x = 8.
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Question 90 of 100
If 2x + 5y = 16 and x + 2y = 7, find the value of x.
Explanationx = 7 - 2y. Substituting: 2(7-2y) + 5y = 16 → 14 + y = 16 → y = 2, so x = 3.
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Question 91 of 100
If x - 2y = -1 and 3x + 4y = 17, find the value of x.
Explanationx = 2y - 1. Substituting: 3(2y-1) + 4y = 17 → 10y = 20 → y = 2, so x = 3.
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Question 92 of 100
Solve for x: 3x - 7 = 14
Explanation3x = 14 + 7 = 21, so x = 21 / 3 = 7.
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Question 93 of 100
Solve for x: 7(x - 1) = 3(x + 5)
Explanation7x - 7 = 3x + 15 → 4x = 22 → x = 5.5.
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Question 94 of 100
Solve for x: 7x - 3 = 4x + 15
Explanation7x - 4x = 15 + 3 → 3x = 18 → x = 6.
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Question 95 of 100
If a + b = 7 and ab = 10, find a² + b².
Explanation(a+b)² = a² + 2ab + b² → 49 = a² + b² + 20 → a² + b² = 29.
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Question 96 of 100
Solve: x² - 8x + 15 = 0
ExplanationFactoring: (x-3)(x-5) = 0, so x = 3 or x = 5.
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Question 97 of 100
Using the previous scenario (father is 3 times the son's age, ages sum to 48), find the father's age.
ExplanationSon's age = 12 (from x + 3x = 48), so father's age = 3 × 12 = 36.
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Question 98 of 100
Find the smaller of two consecutive odd integers whose sum is 40.
ExplanationLet the integers be x and x+2. x + (x+2) = 40 → 2x = 38 → x = 19.
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Question 99 of 100
Solve for x: 3(2x - 1) = 4(x + 2)
Explanation6x - 3 = 4x + 8 → 2x = 11 → x = 5.5.
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Question 100 of 100
If the roots of x² + kx + 12 = 0 are 3 and 4, find the value of k.
ExplanationSum of roots = 3 + 4 = 7 = -k/1, so k = -7. (Check: product = 3×4 = 12 ✓)
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