Yusuf Sila: Algebra

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Showing posts with label Algebra. Show all posts

Problem Set 2 : Equations and Inequalities with Two Variables

4:48 AM

Muhammad Yusuf , Posted in Algebra , 0 Comments

1. Maximize:

P = 3x + 4y

Subject to:

2x + y ≤ 6

x + y ≤ 4

x ≥ 0

y ≥ 0

2. Minimize:

C = 2x - 3y

Subject to:

4x + 5y ≤ 40

2x - y ≥ 0

x ≤ 6

x ≥ 0

y ≥ 0

3. A tailor has 80 square metres of cotton material and 120 square metres of wool. A suit requires 1 square metre of cotton and 3 square metres of wool. A dress requires 2 square metre of cotton and 1 square metre of wool. How many of each should the tailor make to maximize revenue, if a suit sells for $110 and a dress sells for $80?

4. A company makes two types of calculators, Calculator A and Calculator B. Each calculator must be tested after it is assembled. The amount of time required for assembling Calculator A is 4 hours and the amount of time required for assembling Calculator B is also 4 hours. The amount of time for testing Calculator A is 2.5 hours, and the amount of time for testing Calculator B is 1.5 hours. Each week there are 104 working hours for assembling and 60 working hours for testing. If the company makes a profit of $4 on each Calculator A and $2.50 on each Calculator B, how many of each should it produce to maximize its weekly profits?

5. Minimize:

C = x - 4y

Subject to:

2x + 3y ≥ 6

x ≤ 8

y ≤ 12

x ≥ 0

y ≥ 0

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Problem Set 1 : Equations and Inequalities with Two Variables

4:45 AM

Muhammad Yusuf , Posted in Algebra , 0 Comments

1. Sketch the inequality x + 2y < 4

2. Sketch the inequality 5x + 2y ≥ -10

3. John and Mary wrote a Math test. Two times John's score was 60 more than Mary's score. Two times Mary's score was 90 more than John's score. Determine their two scores.

4. Sketch the equations 4x - y = 8 and 2x - 3y = 2 on the same graph. Find the point of intersection of the two lines.

5. Sketch the points that satisfy:

3x + y ≥ 12

x - y ≤ 4

2x - 3y ≤ 6

6. Sketch the points that satisfy:

2x + y > 4

x ≥ 2

x + y < 5

7. A company makes cars and trucks. In any given week, a total of up to 400 vehicles can be made. Draw a graph showing the number of cars and trucks that could be made in one week.

8. Michael plans to spend up to 12 hours reviewing Science and Math in preparation for examinations. Michael is not as good in Science as he is in Math, so he wants to study at least two times more for science then he does for math. Draw a graph showing how much time she could spend studying each subject.

9. Find the area of the triangle enclosed by the lines -4x+3y = 5, 3x+2y = 43 and -5x+8y = 19.

10. If the system of equations

px + qy = 8

3x - qy = 38

has the solution (x, y) = (2, -4). Determine p and q.

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Rules for Inequalities

4:43 AM

Muhammad Yusuf , Posted in Algebra , 0 Comments

1. Adding any number to both sides of an inequality preserves the inequality.

If a < b, then a + c < b + c.

2. Multiplying or dividing both sides of an inequality by a positive number preserves the inequality.

If a < b and c > 0, then ac < bc and a/c < b/c .

3. Multiplying both sides of an inequality by a negative number changes the direction of the inequality.

If a < b and c < 0, then ac > bc and a/c > b/c .

4. If 0 < a < b, then a² < b².

5. If 0 < a < b, then 1/a > 1/b .

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Problem Set : Inequalities in One Variable

4:42 AM

Muhammad Yusuf , Posted in Algebra , 0 Comments

1. The average of a set of n integers is 10. If we remove the integer 2 from this set, the average of the remaining integers is 14. What is the value of n?

2. In a bin at the Cayley Convenience Store, there are 200 candies. Of these candies, 90% are black and the rest are gold. After Matilda eats some of the black candies, 80% of the remaining candies in the bin are black. How many black candies did Matilda eat?

3. The five expressions 2x + 1, 2x - 3, x + 2, x + 5 and x - 3 can be arranged in a different order so that the sum of the first three expressions is 4x+3 and the sum of the last three expressions is 4x + 4. What is the middle expression in the new list?

4. Solve 5x - 2 ≤ 3x - 10 and sketch your solution.

5. Solve 10 - 7x < -4x - 9 and sketch your solution.

6. Solve -1/2(2 + 5x) ≥ 2/3(15 - 3x) and sketch your solution.

7. How many integer values of x satisfy 1/3(x – 1) < 5/7 < 1/5(x+4) ?

8. How many positive integers p satisfy -1 < √p - √100 < 1?

9. If -2 < x < 3 then determine a and b in a < 2x + 3 < b.

10. What values of x satisfy the inequality -3 < 5 - 2/x < 3? Sketch your solution.

11. Solve 2 - 1/x < 3 and sketch your solution.

12. Solve 2/x + 3 ≥ 4 and sketch your solution.

13. The front wheel of Georgina's bicycle has a diameter of 0.75 metres. She cycled for 6 minutes at a speed of 24 kilometres per hour. How many complete rotations did the wheel made during this time?

14. A computer software retailer has 1200 copies of a new software package to sell. From past experience, she knows that:

Half of them will sell right away at the original price she sets,
Two-thirds of the remainder will sell later when the price is reduced by 40%, and
The remaining copies will sell in a clearance sale at 75% off the original price.

In order to make a reasonable profit, the total sales revenue must be greater than or equal to $72 000. To the nearest cent, what is the smallest original price she should set?

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Arithmetic Sequences and Linear Functions

1:39 AM

Muhammad Yusuf , Posted in Algebra , 1 Comment

Problem: Consider the diagrams below. If the pattern continues, how many squares will there be in Diagram 50? Diagram 100?

Figure 1 - A sequence of L-shaped square blocks.

In solving problems, it is important to present data in which we can easily see patterns. Table 1 shows the relationship between the diagram numbers and thenumber of squares.

Table 1 – The relationship between the diagram number and the number of squares.

We can solve this problem by “brute force” extending the table up to Figure 100, but that is not very “mathematical.” What mathematics had taught us is to find patterns, and, if possible, make generalizations. Using the first term, theconstant difference and the diagram number, we can form a numerical expression that when simplified will result to the number of squares as shown in Table 2. Looking at the table, we can say that the first term is 3, and thedifference is 2. Using this pattern, it is now easy to compute the number ofsquares in any diagram number.

Examine the table and see if you can find the pattern before proceeding.

Table 2 - Numerical expressions describing the number of squares in each diagram number.

In Table 2, we can see that in the numerical expression column, the constantdifference 2 and the first term 3 appear in every term. The changing quantity (variable) is the figure number - 1. Using the pattern, it is easy to see that the 50^th term is 2(50-1) + 3 = 101 and the 100^th term is 2(100-1) + 3 = 201. In general, Figure n will have 2(n-1) + 3 = 2n + 1 squares.

Table 3 – Generalized expression describing the number ofsquares.

Let us denote the nth term of a sequence by t_n. Since 2 and 3 are constants, if we let a be the first term of the sequence and d be the constant difference, then the formula that will describe the nth term of the sequence is

t_n = a(n-1) + d

Arithmetic Sequence as a Linear Function

Figure 2 shows the graph of the arithmetic sequence and its trend line denoted by the dashed line. Since we have a constant difference, we have a linear function. If we want to get the equation of the linear function that describes the relationship in our problem, since several ordered pairs are given, we can use the slope intercept formula.

Figure 2 – Graph of the d(n) = 2n + 1, where d(n) is the diagram number

If we extend the trend line, it will pass the (0,1) (Why?). Getting (1,3) as our second point, the slope m will be (3-1)/(1-0) = 2. Hence, the equation of our line will be y = 2x + 1 which is of the same form as t_n = 2n + 1 in Table 3.

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