Equation Problems (Perimeter)

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Question 1 of 3

1. Question
Find $x$ $x$ if the perimeter of the triangle below is $44$ $44$ cm.
- $x =$ $x=$ (8)
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Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

First, form an equation knowing that the Perimeter of a shape is the sum of all sides.

$P = 44$ $P=44$ cm

Side $1 = x$ $1=x$ cm

Side $2 = (x + 6)$ $2=(x+6)$ cm

Side $3 = (3 x - 2)$ $3=(3x-2)$ cm

$P$ $P$ $=$ $=$ Side $1$ $1$ $+$ $+$ Side $2$ $2$ $+$ $+$ Side $3$ $3$

$44$ $44$ $=$ $=$ $x$ $x$ $+$ $+$ $(x + 6)$ $(x+6)$ $+$ $+$ $(3 x - 2)$ $(3x-2)$ Substitute the values

$44$ $44$ $=$ $=$ $5 x + 4$ $5x+4$ Simplify

$5 x + 4$ $5x+4$ $=$ $=$ $44$ $44$

To solve for $x$ $x$ , it needs to be alone on one side.

Start by moving $4$ $4$ to the other side by subtracting $4$ $4$ from both sides of the equation.

$5$ $5$ $x$ $x$ $+ 4$ $+4$ $=$ $=$ $44$ $44$

$5$ $5$ $x$ $x$ $+ 4$ $+4$ $- 4$ $-4$ $=$ $=$ $44$ $44$ $- 4$ $-4$

$5$ $5$ $x$ $x$ $=$ $=$ $40$ $40$ $4 - 4$ $4-4$ cancels out

Finally, remove $5$ $5$ by dividing both sides of the equation by $5$ $5$ .

$5$ $5$ $x$ $x$ $=$ $=$ $40$ $40$

$5$ $5$ $x$ $x$ $\div 5$ $divide5$ $=$ $=$ $40$ $40$ $\div 5$ $divide5$

$x$ $x$ $=$ $=$ $8$ $8$ $5 \div 5$ $5divide5$ cancels out

$x = 8$ $x=8$
Question 2 of 3

2. Question
An isosceles triangle has a perimeter of $55$ $55$ cm and a base that is $5$ $5$ cm less than either of the other $2$ $2$ equal sides. What are the lengths of all three sides?
- Equal Side $1 =$ $1=$ (20)cm
  
  Equal Side $2 =$ $2=$ (20)cm
  
  Base $=$ $=$ (15)cm
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Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

First, label the values and form an equation knowing that the Perimeter of a shape is the sum of all sides.

Equal Side $1 = x$ $1=x$

Equal Side $2 = x$ $2=x$

Base $= (x - 5)$ $=(x-5)$

$P = 55$ $P=55$ cm

Equal Side $1$ $1$ $+$ $+$ Equal Side $2$ $2$ $+$ $+$ Base $=$ $=$ $P$ $P$

$x$ $x$ $+$ $+$ $x$ $x$ $+$ $+$ $x - 5$ $x-5$ $=$ $=$ $55$ $55$ Substitute the values

$3 x - 5$ $3x-5$ $=$ $=$ $55$ $55$ Simplify

To solve for $x$ $x$ , it needs to be alone on one side.

Start by moving $5$ $5$ to the other side by adding $5$ $5$ to both sides of the equation.

$3$ $3$ $x$ $x$ $- 5$ $-5$ $=$ $=$ $55$ $55$

$3$ $3$ $x$ $x$ $- 5$ $-5$ $+ 5$ $+5$ $=$ $=$ $55$ $55$ $+ 5$ $+5$

$3$ $3$ $x$ $x$ $=$ $=$ $60$ $60$ $- 5 + 5$ $-5+5$ cancels out

Finally, remove $3$ $3$ by dividing both sides of the equation by $3$ $3$ .

$3$ $3$ $x$ $x$ $=$ $=$ $60$ $60$

$3$ $3$ $x$ $x$ $\div 3$ $divide3$ $=$ $=$ $60$ $60$ $\div 3$ $divide3$

$x$ $x$ $=$ $=$ $20$ $20$ $3 \div 3$ $3divide3$ cancels out

Substitute $x = 20$ $x=20$ to find each side length.

Equal Side $1$ $1$ $=$ $=$ $x$ $x$

$=$ $=$ $20$ $20$ cm

Equal Side $2$ $2$ $=$ $=$ $x$ $x$

$=$ $=$ $20$ $20$ cm

Base $=$ $=$ $x$ $x$ $- 5$ $-5$

$=$ $=$ $20$ $20$ $- 5$ $-5$

$=$ $=$ $15$ $15$ cm

Check our work

To confirm our answer, add the $3$ $3$ numbers and check if the sum is $55$ $55$ .

$20 + 20 + 15$ $20+20+15$ $=$ $=$ $55$ $55$

$55$ $55$ $=$ $=$ $55$ $55$

Since the equation is true, the answer is correct.

Equal Side $1 = 20$ $1=20$ cm

Equal Side $2 = 20$ $2=20$ cm

Base $= 15$ $=15$ cm
Question 3 of 3

3. Question
A rectangle is twice as long as it is wide. The total perimeter is $126$ $126$ cm. What are the dimensions of this rectangle?
- Length $=$ $=$ (42)cm
  
  Width $=$ $=$ (21) cm
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Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

First, draw a rectangle and label it using the given statement: “A rectangle is twice as long as it is wide.”

This means if the width is $w$ $w$ , the length should be $2 w$ $2w$

Form an equation knowing that the Perimeter of a shape is the sum of all sides.

$P = 126$ $P=126$ cm

Side $1$ $1$ and $2 = w$ $2=w$ cm

Side $3$ $3$ and $4 = 2 w$ $4=2w$ cm

$P$ $P$ $=$ $=$ Side $1$ $1$ $+$ $+$ Side $2$ $2$ $+$ $+$ Side $3$ $3$ $+$ $+$ Side $4$ $4$

$126$ $126$ $=$ $=$ $w$ $w$ $+$ $+$ $w$ $w$ $+$ $+$ $2 w$ $2w$ $+$ $+$ $2 w$ $2w$ Substitute the values

$126$ $126$ $=$ $=$ $6 w$ $6w$ Simplify

$6 w$ $6w$ $=$ $=$ $126$ $126$

Now, to solve for $w$ $w$ , it needs to be alone on one side.

Remove $6$ $6$ by dividing both sides of the equation by $6$ $6$ .

$6$ $6$ $w$ $w$ $=$ $=$ $126$ $126$

$6$ $6$ $w$ $w$ $\div 6$ $divide6$ $=$ $=$ $126$ $126$ $\div 6$ $divide6$

$w$ $w$ $=$ $=$ $21$ $21$ $6 \div 6$ $6divide6$ cancels out

Finally, substitute $w = 21$ $w=21$ to the dimensions of the rectangle.

Length $=$ $=$ $2$ $2$ $w$ $w$

$=$ $=$ $2 ($ $2($ $21$ $21$ $)$ $)$

$=$ $=$ $42$ $42$ cm

Width $=$ $=$ $w$ $w$

$=$ $=$ $21$ $21$ cm

Length $= 42$ $=42$ cm

Width $= 21$ $=21$ cm

Equation Problems (Perimeter)

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Quiz summary

Information

1. Question

Hint

Fantastic!

Inverse Operations

2. Question

Hint

Good Job!

Inverse Operations

3. Question

Hint

Excellent!

Inverse Operations

Quizzes

$P$ $P$	$=$ $=$	Side $1$ $1$ $+$ $+$ Side $2$ $2$ $+$ $+$ Side $3$ $3$
$44$ $44$	$=$ $=$	$x$ $x$ $+$ $+$ $(x + 6)$ $(x+6)$ $+$ $+$ $(3 x - 2)$ $(3x-2)$	Substitute the values
$44$ $44$	$=$ $=$	$5 x + 4$ $5x+4$	Simplify
$5 x + 4$ $5x+4$	$=$ $=$	$44$ $44$

$5$ $5$ $x$ $x$ $+ 4$ $+4$	$=$ $=$	$44$ $44$
$5$ $5$ $x$ $x$ $+ 4$ $+4$ $- 4$ $-4$	$=$ $=$	$44$ $44$ $- 4$ $-4$
$5$ $5$ $x$ $x$	$=$ $=$	$40$ $40$	$4 - 4$ $4-4$ cancels out

Equal Side $1$ $1$ $+$ $+$ Equal Side $2$ $2$ $+$ $+$ Base	$=$ $=$	$P$ $P$
$x$ $x$ $+$ $+$ $x$ $x$ $+$ $+$ $x - 5$ $x-5$	$=$ $=$	$55$ $55$	Substitute the values
$3 x - 5$ $3x-5$	$=$ $=$	$55$ $55$	Simplify

$3$ $3$ $x$ $x$ $- 5$ $-5$	$=$ $=$	$55$ $55$
$3$ $3$ $x$ $x$ $- 5$ $-5$ $+ 5$ $+5$	$=$ $=$	$55$ $55$ $+ 5$ $+5$
$3$ $3$ $x$ $x$	$=$ $=$	$60$ $60$	$- 5 + 5$ $-5+5$ cancels out

Base	$=$ $=$	$x$ $x$ $- 5$ $-5$
	$=$ $=$	$20$ $20$ $- 5$ $-5$
	$=$ $=$	$15$ $15$ cm

$20 + 20 + 15$ $20+20+15$	$=$ $=$	$55$ $55$
$55$ $55$	$=$ $=$	$55$ $55$

$6$ $6$ $w$ $w$	$=$ $=$	$126$ $126$
$6$ $6$ $w$ $w$ $\div 6$ $divide6$	$=$ $=$	$126$ $126$ $\div 6$ $divide6$
$w$ $w$	$=$ $=$	$21$ $21$	$6 \div 6$ $6divide6$ cancels out

Length	$=$ $=$	$2$ $2$ $w$ $w$
	$=$ $=$	$2 ($ $2($ $21$ $21$ $)$ $)$
	$=$ $=$	$42$ $42$ cm