Grade 8 questions on applications of linear equations with solutions and explanations included.
Three times a number increased by ten is equal to twenty less than six times the number. Find the number.
Let the number be \(x\). "Three times a number increased by 10" is \(3x + 10\). "Is equal" is \(=\). "Twenty less than six times the number" is \(6x - 20\). Therefore: \[ 3x + 10 = 6x - 20 \] Solving: \[ 3x - 6x = -20 - 10 \] \[ -3x = -30 \] \[ x = 10 \] Check: \(3 \times 10 + 10 = 40\) and \(6 \times 10 - 20 = 40\).
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.
Let the number be \(x\). "Twice the difference of a number and 3 is added to 4" is \(2(x - 3) + 4\). "The result is" is \(=\). "22 more than four times the number" is \(4x + 22\). Thus: \[ 2(x - 3) + 4 = 4x + 22 \] Solving: \[ 2x - 6 + 4 = 4x + 22 \] \[ 2x - 4x = 22 - 4 + 6 \] \[ -2x = 24 \] \[ x = -12 \]
The sum of two numbers is 64. The difference of the two numbers is 18. What are the numbers?
Let \(x\) be the smaller number. The larger number is \(x + 18\). The sum of the two numbers is: \[ x + (x + 18) = 64 \] \[ 2x + 18 = 64 \] \[ 2x = 46 \] \[ x = 23 \] Larger number: \(x + 18 = 41\).
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.
Let \(W\) be the width. Length: \(L = 2W + 10\). Perimeter formula: \[ 62 = 2L + 2W \] Substitute \(L\): \[ 62 = 2(2W + 10) + 2W \] \[ 62 = 4W + 20 + 2W \] \[ 62 = 6W + 20 \] \[ 6W = 42 \] \[ W = 7 \] Length: \(L = 2(7) + 10 = 24\).
The average of 35, 45 and \( x \) is equal to five more than twice \( x \). Find \( x \).
Average: \[ \frac{35 + 45 + x}{3} = 2x + 5 \] Multiply both sides by 3: \[ 35 + 45 + x = 6x + 15 \] \[ 80 + x = 6x + 15 \] \[ 65 = 5x \] \[ x = 13 \]
The difference in the measures of two supplementary angles is \( 102^{\circ} \). Find the two angles.
Let the smaller angle be \(y\). Then the larger angle is \(y + 102^\circ\). Supplementary angles sum to \(180^\circ\): \[ y + (y + 102) = 180 \] \[ 2y + 102 = 180 \] \[ 2y = 78 \] \[ y = 39 \] Larger angle: \(39 + 102 = 141^\circ\).
Two complementary angles are such that one is \( 14^{\circ} \) more than three times the second angle. What is the measure of the larger angle.
Let the smaller angle be \(y\). Larger angle: \(3y + 14^\circ\). Complementary angles sum to \(90^\circ\): \[ 3y + 14 + y = 90 \] \[ 4y = 76 \] \[ y = 19 \] Larger angle: \(3(19) + 14 = 71^\circ\).
The sum of a positive even integer number and the next third even integer is equal to 150. Find the number.
Let \(x\) be the even integer. The third next even integer is \(x + 6\). Sum: \[ x + (x + 6) = 150 \] \[ 2x + 6 = 150 \] \[ 2x = 144 \] \[ x = 72 \]
The average of three odd successive numbers is equal to 129. What is the largest of the three numbers?
Let the numbers be \(x, x+2, x+4\). Average: \[ \frac{x + (x+2) + (x+4)}{3} = 129 \] \[ \frac{3x + 6}{3} = 129 \] \[ 3x + 6 = 387 \] \[ 3x = 381 \] \[ x = 127 \] Largest: \(127 + 4 = 131\).
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?
Let the smaller be \(x\), then the larger is \(x + 42\). Average: \[ \frac{x + (x + 42)}{2} = 40 \] \[ \frac{2x + 42}{2} = 40 \] \[ 2x + 42 = 80 \] \[ 2x = 38 \] \[ x = 19 \] Numbers: \(19\) and \(61\).