Intermediate Algebra Problems With Answers -
Sample 3

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

  1. Solve the follwoing system of equations

    -x/2 + y/3 = 0

    x + 6y = 16

  2. How many real solutions does each quadratic equation shown below have?

    a) x 2 + (4 / 5) x = - 1/4

    b) x 2 - 7x + 10 = 0

    c) x 2 - (2/3) x + 1/9= 0

  3. Does the data in the table below represents y as a function of x? Explain.

    x y
    1 - 10 4
    11 - 20 6
    21 - 30 8
    30 - 40 12
    40 - 50 16
    51 - 60 34



  4. Solve the following quadratic equation.

    0.01 x 2 - 0.1 x - 0.3 = 0


  5. In a cafeteria, 3 coffees and 4 donuts cost $10.05. In the same cafeteria, 5 coffees and 7 donuts cost $17.15 Dirhams. How much do you have to pay for 4 coffees and 6 donuts?


  6. Find the slope of the lines through the given points and state whether each line is vertical, horizontal or neither.

    Line L1 : (-2 , 3) and (8 , 3)

    Line L2 : (4 , 3) and (4 , -3)

    Line L3 : (-1 , 7) and (3 , -3)

  7. Find four consecutive even integer numbers whose sum is 388.


  8. Going for a long trip, Thomas drove for 2 hours and had lunch. After lunch he drove for 3 more hours at a speed that is 20 km/hour more than before lunch. The total trip was 460 km.

    a) What was his speed after lunch?


  9. Compare the following expressions: a) 2 -4 , b) (-2) -4 , c) (-1/2) 4


  10. Evaluate and convert to scientific notation.

    (3.4 10 11)(5.4 10 -3)

Answers to the Above Questions

  1. We first multiply all terms of the first equation by the LCM of 2 and 3 which is 6.

    6(-x/2 + y/3) = 6(0)

    x + 6y = 16

    We then solve the following equivalent system of equations.

    -3x + 2y = 0

    x + 6y = 16

    which gives the solution

    x = 8/5 and y = 12/5

  2. Find the discriminant of each equation.

    a) (4/5) 2 - 4(1)(1/4) = 16/25 - 1 < 0 , no real solutions.

    b) (-7) 2 - 4(1)(10) = 49 - 40 = 9 > 0 , 2 real solutions.

    c) (-2/3) 2 - 4(1)(1/9) = 4/9 - 4/9 = 0 , 1 real solutions.

  3. No y is not a function of x. According to the table, for x = 30 or x = 40 there are two possible values for the output.



  4. Multiply all terms of the equation by 100 to obtain an equivalent equation with integer coefficients.

    x 2 - 10 x - 30 = 0

    Discrminant = (-10) 2 - 4(1)(-30) = 220

    Solutions: x = (10 ~+mn~ 2√55) / 2 = 5 ~+mn~ √55

  5. Let x be the price of 1 coffee and y be the price of 1 donut.

    We now use "3 coffees and 4 donuts cost $10.05" to write the equation

    3x + 4y = 10.05

    and use "5 coffees and 7 donuts costs $17.15 " to write the equation

    5x + 7y = 17.15

    Subtract the terms of the first equation from the terms of the second equation to obtain

    2x + 3y = 7.10

    Mutliply all terms of the last equation to obtain

    4x + 6y = 14.2

    4 coffees and 6 donuts cost $14.2.

  6. Slope of line L1 = (3 - 3)/(8 + 2) = 0 , horizontal line.

    Slope of line L2 = (-3 - 3)/(4 - 4) = -6/0 undefined, vertical line.

    Slope of line L3 = (-3 - 7) / (3 + 1) = -10/3, L3 is neither horizontal nor vertical.

  7. Let x, x + 2, x + 4 and x + 6 be the 4 consecutive integer.

    Their sum is 388, hence the equation: x + (x + 2) + (x + 4) + (x + 6) = 388

    Solve the above equation to find x = 94.

    The four numbers are: 94, 96, 98 and 100. Add them to check that their sum is 388.

  8. Let x be the speed before lunch, hence the distance driven before lunch is equal to 2 x. After lunch his speed is 20 km/hr more than before lunch and is therefore x + 20. The distance after lunch is 3 (x + 20).

    The total distance is 460 , hence the equation: 2 x + 3 (x + 20) = 460

    Solve the above equation to find speed before lunch x = 80 km/hr

    The speed after lunch is 20 km/hr more than before lunch and is therefore equal to 80 km/hr + 20 km/hr = 100 km/hr.

  9. We first simplify each expression.

    2 -4 = 1 / 2 4 = 1/16

    (-2) -4 = 1 / (-2) 4 = 1/16

    (-1/2) 4 = (-1) 4 / 2 4 = 1 / 16

    The 3 expression simplify to the same value.

  10. (3.4 10 11)(5.4 10 -3) = (3.4 5.4) 10 11 x 10 -3 = 18.36 10 8 = 1.836 10 9



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