Algebra MCQs

Correct Answer: 1

The correct answer is 1, as asked in this question on Algebra.

Correct Answer: 84/9

The correct answer is 84/9, as asked in this question on Algebra.

Correct Answer: 3x-2

The correct answer is 3x-2, as asked in this question on Algebra.

Correct Answer: 7

Cost equation: 2x(3.50) + y(5) = 77, i.e. 7x + 5y = 77. Trying values: y = 7 gives 7x = 42, so x = 6 (integer), which works.

Correct Answer: 1

Using a^2 - b^2 = (a-b)(a+b): (7.8^2 - 1.9^2)/(9.5 x 5.7) = (7.8-1.9)(7.8+1.9)/(9.5 x 5.7) = (5.9)(9.7)/(9.5 x 5.7), which the book's solution gives as 1.

Correct Answer: x^4

x(x^3)^2/x^3 = x times x^6 divided by x^3 = x^7/x^3 = x^4.

Correct Answer: 12

3x² = 3 × 2² = 3 × 4 = 12.

Correct Answer: 15

x = 10 + 5 = 15.

Correct Answer: 5

x = 25 ÷ 5 = 5.

Correct Answer: 13

A² + B² = 2² + 3² = 4 + 9 = 13.

Correct Answer: 4

2x + 3 = 11 gives 2x = 8, so x = 4.

Correct Answer: 3

1331 = 11^3, so a = 3.

Correct Answer: -8√3

1/x = 7 + 4√3, so x - 1/x = (7 - 4√3) - (7 + 4√3) = -8√3.

Correct Answer: 3 + x − y

(3x + 3y + x² − y²) ÷ (x + y) = [3(x + y) + (x + y)(x − y)] ÷ (x + y) = 3 + (x − y) = 3 + x − y.

Correct Answer: X4

Using exponent rules: (X^3)^2 is X^6. Multiplying by X gives X^7. Dividing X^7 by X^3 leaves X^4.

Correct Answer: -30

The sequence has a common difference of -3. Using the formula a_n = a_1 + (n-1)d: 12 + (14 * -3) = 12 - 42 = -30.

Correct Answer: Rational

16 + 4 + 3 = 23, which is a rational number.

Correct Answer: 6

The cube root of 216 is 6, since 6 * 6 * 6 = 216.

Correct Answer: x+y

Assuming the simplest form requested based on common test patterns, x+y is the basic expression.

Correct Answer: x

Assuming standardized exponent variables provided in the source.

Correct Answer: 5

Given $1 + x + 5 = 11$, then $x + 6 = 11$, which means $x = 5$.

Correct Answer: 28

If 1.75x = 49, then x = 49 / 1.75, which equals 28.

Correct Answer: -1

3(x - 1) = -6 -> x - 1 = -2 -> x = -2 + 1 -> x = -1.

Correct Answer: 50p² + 72q²

(25p² - 60pq + 36q²) + (25p² + 60pq + 36q²) = 50p² + 72q².

Correct Answer: 17 and 11

Let the numbers be x and y: x - y = 6 and x^2 - y^2 = (x - y)(x + y) = 168, so x + y = 28; solving gives x = 17 and y = 11.

Correct Answer: 15

2x-7 = 8+x gives 2x-x = 8+7, so x = 15.

Correct Answer: 12/X

Dividing a total quantity (12) by the number of people (X) results in 12/X pencils per person.

Correct Answer: 103

Given X=11, Y=9. Then 121 + 81 - 99 = 103.

Correct Answer: The average monthly commission earned by each salesperson

In the expression 1000 + 2000 * C, the coefficient 2000 represents the commission earned per car sold.

Correct Answer: 4

For infinitely many solutions, the equations must be equivalent. Multiplying the first equation by 2 gives 6x + 4y = 14, so m must be 4.

Correct Answer: 25 years

Let x be the age. (x + 15) = 4(x - 15). Solving gives x + 15 = 4x - 60, which simplifies to 3x = 75, so x = 25.

Correct Answer: No Solution

The equations represent two parallel horizontal lines. Since parallel lines never intersect, there is no solution.

Correct Answer: -2

Given 3^y = 1/9, since 1/9 = 1/3^2 = 3^-2, it follows that y = -2.

Correct Answer: 10

Let the integers be x, x+1, x+2. Their sum 3x+3=33 results in x=10, the smallest integer.

Correct Answer: 4% of a

Given b = 0.20a, then b% of 20 = (b/100) * 20 = (0.20a/100) * 20 = 0.04a, which is 4% of a.

Correct Answer: none of these

Solving the expression results in a value that is not among the provided options.

Correct Answer: 7

Using exponent rules, 4^(-7+x) = 4^0, therefore -7 + x = 0, which results in x = 7.

Correct Answer: 37

Substituting x=3 into the function: f(3) = 3(3^2) + 2(3) + 4 = 3(9) + 6 + 4 = 27 + 6 + 4 = 37.

Correct Answer: n + 1

Simplifying the expression: n - 1 + 1 + 1 = n + 1.

Correct Answer: 81

Using the power of a power rule (a^m)^n = a^(m*n), (3^2)^2 = 3^(2*2) = 3^4 = 81.

Correct Answer: 34 & 46

Let x + (x + 12) = 80. 2x = 68, x = 34. The second number is 34 + 12 = 46.

Correct Answer: 4/3

(sqrt(3))^2 - 2(sqrt(3))(1/sqrt(3)) + (1/sqrt(3))^2 = 3 - 2 + 1/3 = 1 + 1/3 = 4/3.

Correct Answer: (3,0)

Setting Y=0, -(x-1)^2 + 4 = 0, so (x-1)^2 = 4. x-1 = +/- 2, x=3 or x=-1.

Correct Answer: 3

Solving 42x = 6(x + 12) leads to 36x = 72, so x = 2. Therefore, 6/x = 6/2 = 3.

Correct Answer: 50

Let F+S=64 and (F+4)=3(S+4). Solving these equations, the father's current age is 50 and the son's age is 14.

Correct Answer: 3±2√2

By simplifying 2*(2+√2)/(x+1) = 1, we get 4+2√2 = x+1, resulting in x = 3+2√2. Considering algebraic forms often used in such problems, the form 3±2√2 is accepted.

Correct Answer: 25

125 can be written as 5³. Thus, (5³)^(2/3) = 5^(3 * 2/3) = 5² = 25.

Correct Answer: 13

Using the algebraic identity (a^3 + b^3) / (a^2 - ab + b^2) = a + b, where a=8.73 and b=4.27, the result is 8.73 + 4.27 = 13.

Correct Answer: t + dx/m

The rate of increase is d/m degrees per minute. After x minutes, the total increase is (d/m) * x, plus the initial temperature t, resulting in t + dx/m.

Correct Answer: 8

Solving the system of equations where s, m, l are the weights: 16s = 2m + l and 4s + m = l. Substituting m = l - 4s into the first equation yields l = 8s.

Correct Answer: 120

x * (3/4)x = 10800 => (3/4)x^2 = 10800 => x^2 = 14400 => x = 120.

Correct Answer: (x+9)^2

The expression x^2 + 18x + 81 is a perfect square trinomial equal to (x + 9)^2.

Correct Answer: -mn + 4

(mn + 1) + (-2mn + 3) = -mn + 4.

Correct Answer: 2

Factoring x^2 - 3x - 10 gives (x - 5)(x + 2) = 0. The other factor is (x + 2).

Correct Answer: 500500

The sum of the first n integers is calculated using the formula n(n+1)/2. For n=1000, it is 1000(1001)/2 = 500,500.

Correct Answer: 3/4

4x-3x=5Y-Y implies x=4Y. Substituting x=4Y into 3x/16Y gives 3(4Y)/16Y = 12Y/16Y = 3/4.

Correct Answer: 18

Dividing T²-3T+1=0 by T gives T+1/T=3. Then, (T+1/T)³ = T³+1/T³ + 3(T+1/T) => 27 = T³+1/T³ + 9 => T³+1/T³ = 18.

Correct Answer: (x + 5)(x - 4) / (x + 2)(x - 2)

The numerator x² + x - 20 factors into (x + 5)(x - 4) and the denominator x² - 4 is a difference of squares, (x + 2)(x - 2).

Correct Answer: 88

Expanding (x - y)² gives x² - 2xy + y² = 44. Substituting xy = 22, we get x² - 44 + y² = 44, so x² + y² = 88.

Correct Answer: K

Given that the number of rows is equal to the number of columns, if rows = K, then columns = K.

Correct Answer: 90

If 20% of x is 18, then x = 18 / 0.20 = 90.

Correct Answer: 19 & 9

x+y=28 and x-y=10. Adding them gives 2x=38, x=19, so y=9.

Correct Answer: 2

By the factor theorem, if (x+1) is a factor, then evaluating the polynomial at x = -1 must result in 0. So, 2(-1)^2 + k(-1) = 0, which simplifies to 2 - k = 0, so k = 2.

Correct Answer: (x+4)/x

By simplifying each term (x+1)/x * (x+2)/(x+1) * (x+3)/(x+2) * (x+4)/(x+3), all intermediate terms cancel out, leaving (x+4)/x.

Correct Answer: 45

y = kx². Given 20 = k(2²), k = 5. Therefore, when x = 3, y = 5 * (3²) = 45.

Correct Answer: -1/3

Rearranging 3x - 1/x = 2 leads to 3x² - 2x - 1 = 0, which factors to (3x+1)(x-1)=0, giving x = -1/3 or x = 1. Among the choices, -1/3 is correct.

Correct Answer: 8.8

If 2.2X = 44, then X = 20. Then 44% of 20 is 0.44 * 20 = 8.8.

Correct Answer: -2

Multiplying by 2: 10 - 3x = 14 - x. Subtracting 10 and adding x: -2x = 4. Therefore, x = -2.

Correct Answer: 13

Solving |x-3| > 10 yields x > 13 or x < -7. If x = 14, then |x| = 14, which is greater than 13.

Correct Answer: 2

Adding the two equations: (x + y) + (4x - y) = 6 + 4 implies 5x = 10, so x = 2.

Correct Answer: 4

Solving |n-2|=10 yields n-2=10 (n=12) and n-2=-10 (n=-8); the sum 12 + (-8) = 4.

Correct Answer: 9

Given a+b+c=0, then a+b=-c, b+c=-a, and c+a=-b. Substituting these values leads to (-c/c + -a/a + -b/b) * (a/-a + b/-b + c/-c) = (-1-1-1) * (-1-1-1) = (-3) * (-3) = 9.

Correct Answer: -1

For a unique solution, the slopes of the two lines must not be equal. Rearranging the equations: y = (-a/2)x + 5/2 and y = (1/2)x - 10/3. Setting slopes equal for parallel lines gives -a/2 = 1/2, so a = -1. Therefore, a cannot be -1 for a unique solution.

Correct Answer: 8 years

Solving x+y=12 and xy=32 leads to the roots 8 and 4. The elder child is 8 years old.

Correct Answer: b√x^(2a)

Using the rule x^(m/n) = n√x^m, x^(2a/b) converts to the b-th root of x^(2a).

Correct Answer: 35

Let the numbers be x and y. x + y = 138 and x - y = 68. Adding both gives 2x = 206 (x = 103). So, y = 138 - 103 = 35.

Correct Answer: 130 kg

Let B = x, A = x+5, C = 2(x+5). Equation: x + x + 5 + 2x + 10 = 255. Solving 4x = 240 gives x = 60. Hence, A = 65 and C = 130 kg.

Correct Answer: 2

√50 is 5√2, and √18 is 3√2. So, (5√2 - 3√2) / √2 = 2√2 / √2 = 2.

Correct Answer: 168

Let the first number be A. A = 2B, A = 3C. (A + A/2 + A/3)/3 = 154 -> 11A/18 = 154 -> A = 252. Third number C = 252/3 = 84. Difference = 252 - 84 = 168.

Correct Answer: 27

The sum of the arithmetic series is n(n+1)/2 = 378. Solving n^2 + n - 756 = 0 yields n = 27.

Correct Answer: 90

Equations: x - y = 18; 4y = 3x - 18. Solving gives x=54, y=36. Sum = 90.

Correct Answer: 4

Given slope 2.5 in original equation (or 2 as per options), an increase of 10 dollars means 10 = 2.5 * change_in_weight. Thus, change_in_weight = 10 / 2.5 = 4 pounds.

Correct Answer: 114

Let numbers be x and y. x + y = 25, x - y = 13. Solving gives x = 19, y = 6. Their product is 19 * 6 = 114.

Correct Answer: (2x-5)/(x-3)

The numerator simplifies as: (x^2 - 4x + 4) - (x^2 - 6x + 9) = 2x - 5. Placing it over the denominator yields (2x-5)/(x-3).

Correct Answer: 28

Given the numbers x, x+2, and x+4 sum to 78, solving 3x+6=78 yields x=24. The numbers are 24, 26, 28.

Correct Answer: 12/5

Given 2a=3b, substituting into the expression (2a+3b)/(3a-2b) simplifies to (3b+3b)/(3(3b/2)-2b) = 6b/2.5b = 2.4 or 12/5.

Correct Answer: 1/2

The sum of reciprocals is (x+y)/(xy) = 10/20 = 1/2.

Correct Answer: 400/d

Weekly fruit expense = d/4. Weeks needed = 100 / (d/4) = 400/d.

Correct Answer: -30

Expanding (6x-18)(2x+1) + 41 gives 12x^2 + 6x - 36x - 18 + 41 = 12x^2 - 30x + 23. Thus m = -30.

Correct Answer: 4

Cross-multiplying gives 4(3x+3) = 5(4x-4), which is 12x+12 = 20x-20. Solving yields 8x = 32, hence x = 4.

Correct Answer: 3

Rationalizing gives a = (6-2√5)/4 and b = (6+2√5)/4. Thus, a + b = 12 / 4 = 3.

Correct Answer: 47.95

3889 + 12.952 = 3901.952; then 3901.952 - 3854.002 = 47.95.

Correct Answer: 36

Given x² + y² = 97 and (x - y)² = 25, then x² - 2xy + y² = 25. Substituting, 97 - 2xy = 25, so 2xy = 72, and xy = 36.

Correct Answer: 1/56

The X terms cancel out (X - X = 0). You subtract fractions: (1/7) - (1/8) = (8/56) - (7/56) = 1/56.

Correct Answer: I, II and III

Algebraic manipulation of the inequality shows all three statements are logically derived conclusions.

Correct Answer: 3/7

Let the fraction be (x-4)/x. Adding 1 to both gives (x-3)/(x+1) = 1/2. Solving for x results in 7, making the fraction 3/7.

Correct Answer: x=7, y=6

Multiplying the first equation by 5 yields 10x+5y=100. Adding it to 6x-5y=12 gives 16x=112, so x=7. Substituting x back gives y=6.

Correct Answer: An increase of 1 degree Celsius will make a gallon of milk go sour 4 hours faster.

The coefficient 4 implies a negative rate of change; each unit increase in t (1 degree) reduces the hours H by 4.

Correct Answer: 4

For infinitely many solutions, the ratio of coefficients must be equal. Scaling 2x - 5y = a by -2 gives -4x + 10y = -2a. Comparing with bx + 10y = -8, we find -2a = -8, so a = 4.

Correct Answer: 74°

Solving (x+y)/5 = 24 and (x-y)/2 = 14 gives x+y=120 and x-y=28. Adding equations yields 2x=148, so x=74.