Intermediate Algebra Problems With Answers -
Sample 1

A set of intermediate algebra problems, with answers, are presented. The solutions are at the bottom of the page.

  1. For what value of the constant K does the equation 2x + K x = 3 have one solution?.

  2. For what value of the constant K does the equation K x2 + 2x = 1 have two real solutions?

  3. For what value of the constant K does the system of equations 2x - y = 4 and 6x - 3y = 3K have an infinite number of solutions?

  4. Find Y so that the points A(0 , 0), B(2 , 2) and C(-4 , Y) are the vertices of a right triangle with hypotenuse AC.

  5. Find M so that the lines with equations -2x + My = 5 and 4y + x = -9 are perpendicular.

  6. The length of a rectangular field is 7/5 its width. If the perimeter of the field is 240 meters, what are the length and width of the field?

  7. Peter wants to get $200,000 for his house. An agent charges 20% of the selling price for selling the house for Peter.
    a) What should be the selling price?
    b) What will be the agent's commission?

  8. John's annual salary after a raise of 15% is $45,000. What was his salary before the raise?

  9. It took Malcom 3.5 hours to drive from city A to city B. On his way back to city A, he increased his speed by 20 km per hour and it took him 3 hours.
    a) Find the average speed for the whole journey.

  10. Let R be a relation defined by R = {(4,3),(x 2,2),(1,6),(-4,0)}.
    a) Find all values of x so that R is not a function.

Answers to the Above Questions
  1. Solve for x to obtain: x = 3/(2 + k)
    The given equation has one solution for all real values of k not equal to -2.

  2. Rewrite the given quadratic equation in standard form: Kx 2 + 2x - 1 = 0
    Discriminant = 4 - 4(K)(-1) = 4 + 4K
    For the equation to have two real solutions, the discriminant has to be positive. Hence we need to solve the inequality 4 + 4K > 0.
    The solution set to the above inequality is given by: K > -1 for which the given equation has two real solutions.

  3. For the system of equations 2x - y = 4 and 6x - 3y = 3 K to have an infinite number of solutions, the two equations must be equivalent.
    If we multiply all terms of the first equation by 3, we obtain
    6x - 3y = 12
    For the two equations to be equivalent we must have 12 = 3K which when solved gives K = 4.

  4. Let us first find the square of the lengths of the sides of triangle A, B and C.
    Hypotenuse: AC 2 = 16 + y 2
    Side AB: AB 2 = 4 + 4 = 8
    Side BC: BC 2 = (y - 2) 2 + 36
    We now apply Pythagora's theorem: 16 + y 2 = 8 + (y - 2) 2 + 36
    Solve the equation to find y = 8.
  5. We first find the slopes of the two lines: 2/M and -1/4
    For two lines to be perpendicular, the product of their slopes must be equal to -1. Hence the equation (2/M)*(-1/4) = -1. Solve for M to find M = 1/2.

  6. Let L be the length and W be the width. L = (7/5)W
    Perimeter: 2L + 2W = 240, 2(7/5)W + 2W = 240
    Solve the above equation to find: W = 50 m and L = 70 m.
  7. Let x be the selling price: x - 20%x = 200,000
    a) Solve for x to find x = $250,000
    b) 20% *250,000 = $50,000
  8. Let x be the salary before the increase. Hence x + 15% x = $45,000
    Solve for x to find x = $39,130
  9. Let x and x + 20 be the speeds of the car from A to B and then from B to A. Hence the distance from A to B may expressed as 3.5 x and the distance from B to A as 3(x + 20)
    The average speed = total distance / total time = (3.5 x + 3 (x + 20)) / (3.5 + 3)
    The distance from A to B is equal to the distance from B to A, hence: 3.5 x = 3(x + 20). Solve for x to obtain x = 120 km/hr.
    We now substitute x by 120 in the formula for the average speed to obtain.
    average speed = 129.2 km/hr
  10. Relation R is not a function if x 2 = 4 or x 2 = 1
    Hence, R is not a function fo the following values of x: -2, -1, 1 and 2.

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