# An equation about a rectangle with given perimeter

I am doing a revision calculator paper and am stuck on an algebra question.

There is a picture of a rectangle. One side is $x-2,$ another side is $2x +1.$

The question is. Setup and solve an equation to work out the value of $x.$

The perimeter of this rectangle is $43$cm.

How do I do this? Sorry I am useless with algebra, and its worth 5 marks. Thanks

• Are you given any other information? Maybe the area of the rectangle, perimeter length of the rectangle or length of the diagonal of the rectangle? – Dan Rust Jun 8 '13 at 14:34
• Yes sorry, the perimeter of this rectangle is 43cm. – crm Jun 8 '13 at 14:36

For the perimeter of a rectangle, we know that the perimeter is the sum of the lengths of all its sides. In a rectangle, opposite sides are equal in length.

$$\text{Perimeter}\; = 2\times\;\text{length}\; + 2\times\;\text{width}$$

$$2(x - 2)+ 2(2x +1) = \color{blue}{2x} \color{red}{\bf -4} + \color{blue}{4x} + \color{red}{\bf 2} =43$$ $$\color{blue}{6x} \color{red}{\bf - 2} = 43$$ $$6x -2 + {\bf 2} = 43 + {\bf 2}$$ $$6x = 45$$ $${\ Khc 16} \times 6x = {\bf \dfrac 16} \times 45$$ $$x = \dfrac {45}{6} = \dfrac{{\bf 3}\times 15}{{\bf 3} \times 2} =\dfrac {15}{2}$$

$$\text{This gives us}\;\;x = \frac{15}{2} = 7\frac12 = 7.5\;\text{cm}$$

The perimeter is the sum of the 4 sides of the rectangle. Hence $$43=2(x-2)+2(2x+1) \iff 43=2x-4+4x+2 \iff 43=6x-2 \iff 6x=45 \iff x=\frac{45}{6}$$

The perimeter $P$ of the rectangle has length 43, and we also know that the sum of the lengths of each edge is $(x-2)+(x-2)+(2x+1)+(2x+1)$. So, from this we get $P=43$ and $P=(x-2)+(x-2)+(2x+1)+(2x+1)$. So, $$(x-2)+(x-2)+(2x+1)+(2x+1)=43.$$ Can you simplify and solve this equation?

• P= (x-2)+(2x+1)+(x-2)+(2x+1) p= 2(x-2)+2(2x+1) (2x1=2-2=0) + (2x2=4+1=5) x=0+5=5 – crm Jun 8 '13 at 14:50
• Not quite. From the center equation, we can simplify the left hand side to get $6x-2=43$ and so if we rearrange this (by first adding $2$ to both sides, and then diving by 6) we get $x=\frac{45}{6}=7.5$. – Dan Rust Jun 8 '13 at 15:14