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

Solve the follwoing system of equations
x/2 + y/3 = 0
x + 6y = 16

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

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 

Solve the following quadratic equation.
0.01 x^{ 2}  0.1 x  0.3 = 0

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?

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)

Find four consecutive even integer numbers whose sum is 388.

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?

Compare the following expressions: a) 2^{ 4} , b) (2)^{ 4} , c) (1/2)^{ 4}

Evaluate and convert to scientific notation.
(3.4 × 10^{ 11})(5.4 × 10^{ 3)}
Answers to the Above Questions

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

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.

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.

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

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.

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.

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.

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.

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.

(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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