**Mathematical problems — Problems 71 to 80**

The ten problems are presented first, followed by the solutions to the ten problems.

**Problem 71:**

Two lines are on a graph. The first line passes through the point (-4, 13) and has a slope of -3/2. The second line passes through the point (10, 8) and has a slope of 1/2. At what point on the graph do these two lines intersect? *This problem can be solved mathematically without actually drawing the lines on graph paper.*

*Problem 72:*

The sum of three consecutive integers is 45. What are the integers?

*Problem 73:*

The sum of four consecutive integers is 38. What is the largest of the four integers?

*Problem 74:*

The sum of 8 consecutive numbers equals 4. What is the product of these same 8 consecutive numbers?

*Problem 75:*

The product of two consecutive odd numbers is 143. What are the two numbers?

*Problem 76:*

The sum of 4 consecutive odd integers is 64. What are the numbers?

*Problem 77:*

The difference between the cubes of two consecutive integers is 127. What are the two integers?

*Problem 78:*

The sum of 5 consecutive even integers is 100. What is the difference between the largest and smallest of these 5 integers?

*Problem 79:*

The sum of five consecutive multiples of 4 is 20. What is the product of the two largest numbers?

*Problem 80:*

In the sequence of terms beginning with 7 and with a common difference of 5, what is the 10th term in that sequence?

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**Solution to Problem 71:**

- Using the point-slope equation for line 1 yields: (y – 13) = -3/2 (x – (-4))
- Thus, y – 13 = (-3/2 • x) – 6
- And, y = (-3/2 • x) + 7
- Using the point-slope equation for line 2 yields: (y – 8) = 1/2 (x – 10)
- Thus, y – 8 = (1/2 • x) – 5
- And, y = (1/2 • x) + 3
- Since the intersection point will be at the same value for y, these two equations can be set as equal to each other.
- Hence, (-3/2 • x) + 7 = (1/2 • x) + 3
- Simplifying yields: 2x = 4
- Thus, x = 2. With x = 2, then y = 4. The point of intersection of the two points is thus (2, 4).

**Solution to Problem 72:**

- x + (x + 1) + (x + 2) = 45
- 3x + 3 = 45
- 3x = 42
- x = 14
- The integers must be 14, 15, 16

*Solution to Problem 73:*

- x + (x + 1) + (x + 2) + (x + 3) = 38
- 4x + 6 = 38
- 4x = 32, or x + 8
- If 8 is the smallest of the integers, then 11 must be the largest.

*Solution to Problem 74:*

- x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5) + (x + 6) + (x + 7) = 4
- 8x + 28 = 4, or 8x = -24
- So, x = -3
- Since x = -3, then the other seven numbers are -2, -1, 0, 1, 2, 3, and 4.
- With zero as one of the numbers, the product has to be equal to 0 (zero)!

*Solution to Problem 75:*

- x • y = 143, with y being 2 greater than x
- Thus, x • (x + 2) = 143
- So, x
^{2}+ 2x – 143 = 0 - Factoring yields: (x – 11) (x + 13) = 0
- Thus, the solutions are 11 and 13 and -11 and -13.

**Solution to Problem 76:**

- x + (x + 2) + (x + 4) + (x + 6) = 64
- 4x + 12 = 64
- 4x = 52. so x = 13
- Thus, the integers are 13, 15, 17, and 19.

**Solution to Problem 77:**

- x
^{3}– (x – 1)^{3}= 127 - x
^{3}– (x – 1)(x^{2}– 2x +1) = 127 - x
^{3}– (x^{3}– 2x^{2}+ x – x^{2}+ 2x – 1) = 127 - x
^{3}– (x^{3}– 3x^{2}+ 3x – 1) = 127 - 3x
^{2}– 3x + 1 = 127 - 3x
^{2}– 3x -126 = 0 - Factoring yields: (3x – 21)(x + 6) = 0
- Thus, x = 7 and the two numbers must be 6 and 7.
- 343 – 216 = 127

*Solution to Problem 78:*

- x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = 100
- 5x + 20 = 100
- 5x = 80
- So, x = 16. The numbers must be 16, 18, 20, 22, and 24.
- The difference between the largest and smallest of the numbers is 24 – 16 = 8
- In actuality, one doesn’t need to know the specific numbers to determine the difference between the largest and smallest of the 5 integers. The difference between the largest and smallest of 5 consecutive even integers will always be 8, no matter what the numbers are.

*Solution to Problem 79:*

- x + (x + 4) + (x + 8) + (x + 12) + (x + 16) = 20
- 5x + 40 = 20
- 5x = -20
- x = -4
- Thus, the 5 multiples of 4 are -4, 0, 4, 8, and 12. Their sum does equal 20.
- The largest two of the five numbers are 8 and 12. Their product is 96.

*Solution to Problem 70:*

- Remember the equation, a
_{n}= a_{1}+ d(n – 1) , with a_{n}being the nth term of an arithmetic sequence, a_{1}being the first term, and d being the common difference between the terms. - Thus, a
_{10}= 7 + 5(10 – 1) = 7 + 45 = 52 - Double-checking yields, 5 – 12 – 17 – 22 – 27 – 32 – 37 – 42 – 47 – 52.

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