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 + 42) / 2 = 40
    Cross multiply and solve
    2x + 42 = 80
    2x = 80 - 42
    2x = 38
    x = 19
    The two numbers are
    x = 19 and x + 42 = 19 + 42 = 61
    19 and 61

More References and links

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