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

Let W be the width. The length L is 2W + 10. The perimeter formula is 2L + 2W = 62. Substituting L gives 2(2W + 10) + 2W = 62. Simplifying yields 6W + 20 = 62. Solving for W gives W = 7. The width is 7 meters and the length is 24 meters.

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

Supplementary angles sum to 180 degrees. Let the smaller angle be y; the larger angle is y + 102. Setting up the equation: y + (y + 102) = 180. Solving gives 2y = 78, so y = 39. The smaller angle is 39 degrees and the larger is 141 degrees.

Linear Equation Word Problems: Grade 8 Questions & Solutions

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

Let the smaller number be x. Since the difference is 18, the larger number is x + 18. The sum is x + (x + 18) = 64. Solving for x gives 2x = 46, so x = 23. The smaller number is 23 and the larger number is 41.

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

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Let the number be \(x\).

"Three times a number increased by 10" translates to \(3x + 10\).
"Is equal to" translates to \(=\).
"Twenty less than six times the number" translates to \(6x - 20\).

Set up the equation and solve:

\(3x + 10 = 6x - 20\)
\(3x - 6x = -20 - 10\) (Move variables to one side and constants to the other)
\(-3x = -30\)
\(x = 10\) (Divide both sides by -3)

Check: \(3(10) + 10 = 40\) and \(6(10) - 20 = 40\). The solution is correct.

Complex Translation: 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.

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Let the number be \(x\).

"Twice the difference of a number and 3 is added to 4" translates to \(2(x - 3) + 4\).
"The result is" translates to \(=\).
"22 more than four times the number" translates to \(4x + 22\).

Set up the equation and solve:

\(2(x - 3) + 4 = 4x + 22\)
\(2x - 6 + 4 = 4x + 22\) (Distribute the 2)
\(2x - 2 = 4x + 22\) (Combine constants on the left)
\(2x - 4x = 22 + 2\) (Move variables to the left, constants to the right)
\(-2x = 24\)
\(x = -12\)

Sums and Differences: The sum of two numbers is 64. The difference of the two numbers is 18. What are the numbers?

View Step-by-Step Solution

Let \(x\) be the smaller number. Since the difference between the two numbers is 18, the larger number must be \(x + 18\).

Set up the equation using their sum:

\(x + (x + 18) = 64\)
\(2x + 18 = 64\) (Combine like terms)
\(2x = 64 - 18\)
\(2x = 46\)
\(x = 23\) (This is the smaller number)

Find the larger number:

Larger number = \(x + 18 = 23 + 18 =\) \(41\)

Check: \(23 + 41 = 64\) and \(41 - 23 = 18\).

Geometric Application (Perimeter): The length of a rectangle is 10 meters more than twice its width. What are the length and width of the rectangle if its perimeter is 62 meters?

View Step-by-Step Solution

Let \(W\) be the width. The length \(L\) is described as \(2W + 10\).

The formula for the perimeter of a rectangle is \(P = 2L + 2W\).

\(62 = 2L + 2W\)
\(62 = 2(2W + 10) + 2W\) (Substitute the expression for L)
\(62 = 4W + 20 + 2W\) (Distribute the 2)
\(62 = 6W + 20\) (Combine like terms)
\(62 - 20 = 6W\)
\(42 = 6W\)
\(W = 7\) meters

Now, find the length:

\(L = 2(7) + 10 = 14 + 10 =\) \(24\) meters

Averages: The average of 35, 45, and \(x\) is equal to five more than twice \(x\). Find \(x\).

View Step-by-Step Solution

To find the average of three numbers, sum them and divide by 3:

\(\dfrac{35 + 45 + x}{3} = 2x + 5\) (Set up the equation based on the problem statement)
\(35 + 45 + x = 3(2x + 5)\) (Multiply both sides by 3 to remove the fraction)
\(80 + x = 6x + 15\) (Combine numbers and distribute the 3)
\(80 - 15 = 6x - x\) (Group variables and constants)
\(65 = 5x\)
\(x = 13\) (Divide by 5)

Supplementary Angles: The difference in the measures of two supplementary angles is \(102^\circ\). Find the two angles.

View Step-by-Step Solution

Supplementary angles add up to \(180^\circ\). Let the smaller angle be \(y\). Because their difference is \(102^\circ\), the larger angle is \(y + 102^\circ\).

\(y + (y + 102) = 180\)
\(2y + 102 = 180\) (Combine like terms)
\(2y = 180 - 102\)
\(2y = 78\)
\(y = 39^\circ\) (This is the smaller angle)

Find the larger angle:

Larger angle = \(39^\circ + 102^\circ =\) \(141^\circ\)

Complementary Angles: 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?

View Step-by-Step Solution

Complementary angles add up to \(90^\circ\). Let the smaller angle be \(y\). The larger angle is \(3y + 14^\circ\).

\(y + (3y + 14) = 90\)
\(4y + 14 = 90\) (Combine like terms)
\(4y = 90 - 14\)
\(4y = 76\)
\(y = 19^\circ\) (This is the smaller angle)

Find the larger angle:

Larger angle = \(3(19) + 14 = 57 + 14 =\) \(71^\circ\)

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

View Step-by-Step Solution

Let \(x\) be the first even integer. Consecutive even integers skip by 2. The sequence is: \(x\) (1st), \(x + 2\) (next), \(x + 4\) (second next), \(x + 6\) (third next).

\(x + (x + 6) = 150\) (Set up the sum of the first and third next)
\(2x + 6 = 150\)
\(2x = 150 - 6\)
\(2x = 144\)
\(x = 72\)

Consecutive Odd Integers: The average of three successive odd numbers is equal to 129. What is the largest of the three numbers?

View Step-by-Step Solution

Let the three successive odd numbers be \(x\), \(x + 2\), and \(x + 4\). (Odd numbers also skip by 2).

\(\dfrac{x + (x + 2) + (x + 4)}{3} = 129\) (Set up the average equation)
\(\dfrac{3x + 6}{3} = 129\) (Combine terms in the numerator)
\(x + 2 = 129\) (Divide numerator terms by 3)
\(x = 127\) (This is the smallest number)

Find the largest number (\(x + 4\)):

Largest number = \(127 + 4 =\) \(131\)

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

View Step-by-Step Solution

Let the smaller number be \(x\). The larger number is \(x + 42\).

\(\dfrac{x + (x + 42)}{2} = 40\) (Set up the average equation)
\(\dfrac{2x + 42}{2} = 40\) (Combine like terms)
\(x + 21 = 40\) (Divide numerator terms by 2)
\(x = 19\) (This is the smaller number)

Find the larger number:

Larger number = \(19 + 42 =\) \(61\)

The two numbers are 19 and 61.

Grade 8 Linear Equations: Real-Life Applications & Word Problems

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