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.

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.

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

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.

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.

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.

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