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 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 102. Find the two angles.

    Solution

    If the difference of measures of two angles is 102, then
    larger angle = smaller angle + 102
    The sum of the measures of two supplementary angles is equal to 180. Hence
    Larger angle + smaller angle = 180
    or
    smaller angle + 102 + smaller angle = 180
    2 smaller angle = 180 - 102 = 78
    smaller angle = 78 / 2 = 39
    larger angle = smaller angle + 102 = 141

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

    Solution

    There are two angles: a larger one and a smaller one. The larger one is such that
    larger = 3 smaller + 14
    The sum of two angles is 90 . Hence
    larger + smaller = 90
    or
    3 smaller + 14 + smaller = 90
    4 smaller = 90 - 14
    4 smaller = 76
    smaller = 76 / 4 = 19
    larger = 3 smaller + 14 = 3 19 + 14 = 71

  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 129. 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 129. Hence
    (x + x + 2 + x + 4) / 3 = 129
    Rewrite above equation as
    (x + x + 2 + x + 4) / 3 = 129 / 1
    Cross multiply 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 the 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

More References and links

Middle School Maths (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers
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