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.
    

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.