Question

1. the length of a rectangle is 5 less than twice its width. the perimeter is 26 meters. find the dimensions of the rectangle.
2. find two consecutive numbers whose sum is 115.
3. find two consecutive odd numbers whose sum is 144.
4. find two consecutive even numbers whose sum is 126.
5. find three consecutive numbers whose sum is 84.
6. find three consecutive even integers such that the sum of the smallest number and twice the middle number is 20 more than the largest number.

Explanation:

7.

Step1: Define variables

Let the width of the rectangle be $w$. Then the length $l = 2w - 5$. The perimeter formula of a rectangle is $P=2(l + w)$. Given $P = 26$.

Step2: Substitute into perimeter formula

Substitute $l$ and $P$ into the formula: $26=2((2w - 5)+w)$.
First, simplify the expression inside the parentheses: $26 = 2(3w - 5)$.

Step3: Distribute the 2

$26=6w-10$.

Step4: Solve for $w$

Add 10 to both sides: $26 + 10=6w$, so $36 = 6w$. Then divide both sides by 6: $w = 6$.

Step5: Solve for $l$

Substitute $w = 6$ into the length formula $l=2w - 5$. So $l=2\times6 - 5=12 - 5 = 7$.

Step1: Define variables

Let the first - consecutive number be $x$, then the second consecutive number is $x + 1$. Their sum is $x+(x + 1)=115$.

Step2: Simplify the equation

Combine like terms: $2x+1 = 115$.

Step3: Solve for $x$

Subtract 1 from both sides: $2x=115 - 1=114$. Then divide both sides by 2: $x = 57$.

Step4: Find the second number

The second number is $x + 1=57+1 = 58$.

Step1: Define variables

Let the first consecutive odd number be $x$, then the second consecutive odd number is $x + 2$. Their sum is $x+(x + 2)=144$.

Step2: Simplify the equation

Combine like terms: $2x+2 = 144$.

Step3: Solve for $x$

Subtract 2 from both sides: $2x=144 - 2 = 142$. Then divide both sides by 2: $x = 71$.

Step4: Find the second number

The second number is $x + 2=71+2 = 73$.

Answer:

Width is 6 meters, length is 7 meters.

8.