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Solving Geometry Problems Involving System of Equations In Two Variables

Updated on August 6, 2011

Solving Geometry Problems Involving System of Equations In Two Variables


Algebra has also found wide applications in Geometry. Among its common problems are problems involving perimeters and angles. In this hub I included five sample problems involving perimeter and angles.


Sample Problem Number One :

A rectangular garden has a perimeter of 100 meters. The length is four times its width. Find the dimensions of the garden.

Solution :

Let X = width

Let Y = length

P = 2L + 2w

Equation one : 2X + 2Y = 100

Equation Two : Y = 4X

Substitute equation two in equation one :

2X + 2 (4X) = 100

2X + 8X = 100

10X = 100

(1/10) 10X = 100 (1/10)

X = 10 ==è width

4X = 4 (10) = 40 ==è lenth


Sample Problem Two :

The perimeter of a rectangular picture frame is 80 centimeter. Two times the width is equal to the length increased by five. Find its dimensions.

Solution :

Let X = width

Let Y = length

P = 2L + 2W

Equation one : 2X + 2Y = 80

Equation two: 2X = Y + 5

We use elimination method by subtraction to solve for the unknown variables.

2X + 2Y = 80

(-)2X (+) -Y = (-) 5

3Y = 75

(1/3) 3Y = 75 (1/3)

Y = 25 ==è length

To solve for width substitute in equation two

2X = 25 + 5

2X = 30

X = 15 ====è width


Sample Problem Three:

The perimeter of a rectangular flower garden is 120 meters. The length is ten meters greater than its width. Find its dimensions.

Solution :

Let X = width

Let Y = length

P = 2L +2W

Equation one : 2X + 2Y = 120

Equation two : Y = X + 10

Substitute equation two in equation one :

2X + 2 (X + 10 ) = 120

2X + 2X + 20 = 120

4X = 120 – 20

4X = 100

(1/4) 4X = 100 (1/4)

X = 25 ==è width

Y = 25 + 10 = 35 ===è lenth


Sample Problem Number Four :

The sum of the two nonright angles in a right triangle is of course 90 degrees. If twice the first is 40 degrees more than three times the second. Find the measurement of the angles of the right triangle.

Solution :

Let X = first angle

Let Y = second angle

Equation one : X + Y = 90

Equation two : 2X = 3Y + 40

From equation one we derive Y = 90 – X the substitute this in equation two.

2X = 3(90-X) + 40

2X = 270 – 3X + 40

2X + 3X = 270 + 40

5X = 310

(1/5) 5X = 310 (1/5)

X = 62 degrees ===è first angle

Y = 90 – 62 = 28 ===è second angle


Sample Problem Number Five :

Two angles are supplementary. The bigger angle is thrice as large as the smaller angle. Find the measurement of the angles.

Solution :

Let X = smaller angle

Let Y = bigger angle

The two angles are supplementary it means that their sum is equal to 180 degrees.

Equation one : X + Y = 180

Equation two : Y = 3X

Substitute equation two in equation one :

X +3X = 180

4X = 180

(1/4) 4X = 180 (1/4)

X = 45 ==èsmaller angle

Y = 3(45) = 135 ====. Bigger angle

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    • profile image

      catherine 

      4 years ago

      it is very great i really like iit

    • profile image

      John Peterson 

      6 years ago

      Trolololololololol

    • cristina327 profile imageAUTHOR

      Maria Cristina Aquino Santander 

      7 years ago from Manila

      Hi DeBorrah I am glad to hear from you again. Thank you for appreciating this hub. Your visit and comments are much appreciated. You have been really a great encouragement to me. Blessings to you and your family. Best regards.

    • DeBorrah K. Ogans profile image

      DeBorrah K Ogans 

      7 years ago

      Cristina, Nice Geometry lesson... You are so smart intellectually and gifted spiritually! What a double blessing that you also Love the Lord!

      Thank You for sharing, In HIS Love, Grace, Joy, Peace & Blessings!

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