Two Equations Two Unknowns

Introduction to two equations and two unknowns:

Two linear equations in the same two variables (unknowns) are called a pair of linear equations in two variables. The most general form of a pair of linear equations is

a1x+b1y+c1=0

a2x+b2y+c2=0

An example of linear system involves two equations and two unknowns:

x+ y=3 and x - y=2

There are the methods used to solve two unknowns with two equations:

Substitution method

Elimination method

Graphing method

Here, we are going to see the problems on two equations and two unknowns by substitution and elimination method.

Two Equations Two Unknowns-solving

Example problem 1:

Solve for the two variables x and y from the following two equations:

3x+y=5
y+5x=2

Solution:

Here, we have to solve the pair of equations with two unknowns by substitution method.

Step 1: We pick any one of the equations and write one variable in terms of the other.

Let us consider the Equation (1):

3x+y=5

Subtract 3x on both sides of the equation

3x+y-3x=5-3x

y=5-3x------------------------Equation (3)

Step 2: Substitute the value of y in Equation (2). We get

y+5x=2

5-3x+5x=2

5+2x=2

Subtract 5 on both sides

2x=2-5

2x=-3

Divide by 2 on both sides of the equation

2x/2=-3/2

x=-1.5

Step 3: Plugging this value of x in Equation (3), we get

y=5-3x

y=5-3(-1.5)

y=5+4.5

y=9.5

So, the solution of two unknowns is (-1.5, 9.5).

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Two Equations Two Unknowns- by Elimination Method:

Example problem 2: Solve for the two variables x and y from the following two equations:

9x – 4y = 2000----------Equation (1)

7x – 3y = 2000----------Equation (2)

Solution:

We have to solve the pair of equations by Elimination method.

Step 1: Equation (1) is multiplied by 3 and Equation (2)is multiplied by 4 to make the coefficients of y equal. Then we get the equations:

27x – 12y = 6000------Equation (3)

28x – 12y = 8000------Equation (4)

Step 2: Equation (3) is subtracted from Equation (4) to eliminate y, because the coefficients

of y are the same. So, we get

(28x – 27x) – (12y – 12y) = 8000 – 6000

i.e., x = 2000

Step 3: Substituting this value of x in (1), we get

9(2000) – 4y = 2000

i.e., y = 4000

So, the solution of two unknowns is (2000, 4000).