Applications of Linear Equations
Problems with Answers for Grade 8

Solutions and explanations to Grade 8 questions on applications of linear equations .

  1. Three times a number increased by ten is equal to twenty less than six times the number. Find the number.

    Solution

    Let the number be x. "Three times a number increased by 10" is mathematically translated as

    3x + 10

    "is equal" is mathematically translated as

    =

    "Twenty less than six times the number" is mathematically translated as

    6x - 20

    The whole sentence "Three times a number increased by ten is equal to twenty less than six times the number" is translated as

    3x + 10 = 6x - 20

    We now solve the above linear equation to find the number x

    3x - 6x = - 20 - 10

    - 3x = - 30

    x = 10

    Check answer

    Three times a number increased by ten : 3 10 + 10 = 40

    Twenty less than six times the number : 6 10 - 20 = 40

  2. If twice the difference of a number and 3 is added to 4, the result is 22 more than four times the number. Find the number.

    Solution

    Let the number be x. "twice the difference of a number and 3 is added to 4" is mathematically translated as

    2(x - 3) + 4

    "the result is" is mathematically translated as

    =

    "22 more than four times the number" is mathematically translated as

    4x + 22

    "twice the difference of a number and 3 is added to 4, the result is 22 more than four times the number" is mathematically translated as

    2(x - 3) + 4 = 4x + 22

    Solve the equation

    2x - 6 + 4 = 4x + 22

    2x - 4x = 22 - 4 + 6

    -2x = 24

    x = -12

  3. The sum of two numbers of two is 64. The difference of the two numbers is 18. What are the numbers?

    Solution

    Let x be the smaller of the two numbers. Since the difference of the two numbers is 18, then the larger number is

    x + 18

    The sum of the two numbers is 64. Hence

    smaller number + larger number = 64 or x + (x + 18) = 64

    Solve for x

    2x + 18 = 64

    2x = 64 - 18

    2x = 46

    x = 23 , the smallest of the two numbers.

    x + 18 = 23 + 18 = 41 , the largest of the two numbers.

  4. The length of a rectangle is 10 meters more than twice its width. What is the length and width of the rectangle if its perimeter is 62 meters.

    Solution

    Let W be the width of the rectangle. "the length of a rectangle is 10 meters more than twice its width" is translated as

    length = 2 W + 10

    The perimeter of the rectangle is given by

    Perimeter = 2 length + 2 width

    62 = 2 (2 w + 10) + 2 W

    Solve the above equation for W

    62 = 4 W + 20 + 2 W

    62 = 6 W + 20

    62 - 20 = 6 W

    42 = 6 W

    W = 7

    length and width are

    width = W = 7 meters , length = 2 W + 10 = 2 (7) + 10 = 24 meters

  5. The average of 35, 45 and x is equal to five more than twice x. Find x.

    Solution

    The average of 35, 45 and x is given by

    (35 + 45 + x) / 3

    The average is equal to five more than twice x. Hence

    (35 + 45 + x) / 3 = 2x + 5

    The above equation can be written as

    (35 + 45 + x) / 3 = (2x + 5) / 1

    Cross product and solve

    1(35 + 45 + x) = 3(2x + 5)

    35 + 45 + x = 6x + 15

    80 + x = 6x + 15

    80 - 15 = 6x - x

    65 = 5x

    x = 13

  6. The difference in the measures of two supplementary angles is 102o. Find the two angles.

    Solution

    If the difference of measures of two angles is 102o, then

    larger angle = smaller angle + 102o

    The sum of the measures of two supplementary angles is equal to 180o. Hence

    Larger angle + smaller angle = 180o

    or

    smaller angle + 102o + smaller angle = 180o

    2 smaller angle = 180 - 102 = 78o

    smaller angle = 78 / 2 = 39o

    larger angle = smaller angle + 102 = 141o

  7. Two complementary angles are such that one is 14o more than three times the second angle. What is the measure of the larger angle.

    Solution

    There two angles: a larger one and a smaller one. The larger one is such that

    larger = 3 smaller + 14o

    The sum of two angles is 90o . Hence

    larger + smaller = 90o

    or

    3 smaller + 14o + smaller = 90o

    4 smaller = 90 - 14

    4 smaller = 76

    smaller = 76 / 4 = 19o

    larger = 3 smaller + 14o = 3 19 + 14 = 71o

  8. The sum of a positive even integer number and the next third even integer is equal to 150. Find the number.

    Solution

    Let x be the positive even integer. The next three even integers are

    x + 2 , x + 4 , x + 6

    The third even integer is x + 6. The sum of x and x + 6 is 150. Hence

    x + x + 6 = 150

    2x = 150 - 6

    2x = 144

    x = 72

  9. The average of three odd successive numbers is equal to 219. What is the largest of the three numbers?

    Solution

    Three odd successive integers are of the form

    x , x + 2 , x + 4

    Their average is equal to 219. Hence

    (x + x + 2 + x + 4) / 3 = 129

    Rewrite above equation as

    (x + x + 2 + x + 4) / 3 = 129 / 1

    Cross mutliply and solve

    (3x + 6)1 = 129(3) 3x + 6 = 387

    3x = 387 - 6

    3x = 381

    x = 127

    The largest of the three numbers is

    x + 4 = 127 + 4 = 131

  10. Two numbers are such that one number is 42 more that the second number and their average is equal to 40. What are the two numbers?

    Solution

    If x is the smallest number, then largest is.

    x + 42

    The average of x and x + 42 is equal to 40. Hence

    (x + x + 40) / 2 = 40

    Cross multiply and solve

    2x + 40 = 80

    2x = 80 - 40

    2x = 40

    x = 20

    The two numbers are

    x = 20 and x + 40 = 60


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Updated: 9 March 2012 (A Dendane)

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