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Please study
Linear Equations before Linear Equations in Two Variables,
if you have not already done so.

There, we explained solving of the Linear Equations
in one variable with a number of solved Examples.

That knowledge is a prerequisite here.

Word Problems in one variable are covered in
Algebra Word Problems

Here we deal with two variables instead of one.
We explain the method of solving with examples.

We apply this knowledge to solve word problems.
A number of word problems are worked out.

Several word problems are given for
practice in exercise with answers.

Practice makes one perfect.
This is especially true for learning of solving
Algebra Problems (Math Problems).
So, take the exercises seriously and practice solving the problems.

Linear Equations in two variables

The general form of Linear Equation in two variables is ax + by + c = 0,
where a, b, c are real numbers (a ≠ 0, b ≠ 0).

Here there are two unknowns and we need two equations to find them.

The two linear equations in two variables which are satisfied by
by the same pair of values for the variables are called
simultaneous linear equations in two variables.

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Solving Simultaneous Linear Equations in Two variables

We eliminate one of the unknowns, using the two
Linear Equations in Two variables.
Then we get a simple equation in the other unknown
from which we find its value. This value is used
in one of the Two Linear Equations in Two variables
to find the other unknown.

The steps involved in the method of solving
simultaneous Linear Equations in Two Variables
may be listed as follows :

Step 1 :

Multiply the given two Linear Equations in Two Variables
by suitable numbers so as to make
the coefficients of one of the unknowns, numerically equal.

Step 2 :

If the numerically equal coefficients are opposite in sign,
add the new equations. Otherwise subtract them.

Step 3 :

The resulting equation is linear in one unknown.
Solve it to obtain the value of one of the unknowns.

Step 4 :

Substitute the value of this unknown in any of the given
two Linear Equations in Two Variables.
Solve it to get the value of the other unknown.

Step 5 :

Check the validity of the Answers by substituting
the values in the given equations.

Let us see an example.

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Example 1 :

Solve the following Equations in Two Variables
2x + 3y -23 = 0, 5x - 8y - 20 = 0.

Solving the two given Linear Equations in Two Variables :

The given equations are:
2x + 3y = 23...........................................(i)
5x - 8y = 20 ...........................................(ii)
To solve let us make the coefficients of y same.
For this we have to multiply (i) with 8 and (ii) with 3.
(i) x 8 gives 8 x 2x + 8 x 3y = 8 x 23 ⇒ 16x + 24y = 184.........(iii)
(ii) x 3 gives 3 x 5x - 3 x 8y = 3 x 20 ⇒ 15x - 24y = 60.........(iv)
Here numerically equal coefficients of y are opposite in sign
∴ We have to add the equations (iii) and (iv).
(iii) + (iv) gives 16x + 15x = 184 + 60 ⇒ 31x = 244
x = 244⁄31
Using this in Equation (i), we get
2(244⁄31) + 3y = 23
Multiplying throughout with 31, we get
2(244) + 31 x 3y = 31 x 23 ⇒ 488 + 93y = 713
⇒ 93y = 713 - 488 = 225 ⇒ y = 225⁄93 = 75⁄31
Thus x = 244⁄31 and y = 75⁄31. Ans.

Substitute these values in the given equations and
verify whether L.H.S. and R.H.S. are equal.

L.H.S. of (i) = 2x + 3y = 2(244⁄31) + 3(75⁄31)
= {2(244) + 3(75)}⁄31 = 713⁄31 = 23 = R.H.S. of (i) (verified.)

L.H.S. of (ii) = 5x - 8y = 5(244⁄31) - 8(75⁄31)
= {5(244) - 8(75)}⁄31 = 620⁄31 = 20 = R.H.S. of (ii) (verified.)

You will see more problems when you see the solving
of the Linear equations in two variables formed in
the Word Problems in the Solved Examples given below.

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Word Problems involving Equations in two variables

The application of the knowledge of solving
the Equations in two variables learned
above is found in solving the word problems
given in the following Links.

Set 1 of Word Problems

Set 2 of Word Problems

Set 3 of Word Problems

Set 4 of Word Problems

Exercise : Word Problems

The Sets of Word Problems given above
contain Exercise problems also.
Solving those problems require the
knowledge, you learned here.
So, please attempt those Exercise problems.

Progressive Learning of Math : Linear equations in two variables

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