Question

1. the measure of one angle is 8 degrees more than one - third the measure of another. if the angles are supplementary, find both angle measures.
2. find three consecutive odd integers such that the sum of the smallest number and four times the largest number is fifty - one more than twice the median number.

Explanation:

Step1: Define the consecutive odd integers

Let the three consecutive odd integers be $x$, $x + 2$, $x+4$.

Step2: Set up the equation based on the problem - statement

The sum of the smallest and four times the largest is fifty - one more than twice the median number. So, $x+4(x + 4)=2(x + 2)+51$.

Step3: Expand the equation

Expand both sides: $x+4x+16 = 2x+4 + 51$.

Step4: Combine like - terms

On the left - hand side, $x+4x=5x$, so we have $5x+16$. On the right - hand side, $4 + 51=55$, so we have $2x+55$. The equation becomes $5x+16=2x+55$.

Step5: Solve for $x$

Subtract $2x$ from both sides: $5x-2x+16=2x-2x + 55$, which simplifies to $3x+16=55$. Then subtract 16 from both sides: $3x+16 - 16=55 - 16$, giving $3x=39$. Divide both sides by 3: $x = 13$.

Step6: Find the three consecutive odd integers

The three consecutive odd integers are $x=13$, $x + 2=15$, $x+4=17$.

Answer:

13, 15, 17