# SOLVING PAIRS OF LINEAR EQUATIONS BY SUBSTITUTION EXAMPLES

Solving Pairs of Linear Equations by Substitution Examples :

In this section, we will learn, how to solve linear equations by substitution method.

We use the following steps to solve a linear equations in substitution method.

## Substitution Method

Step 1 :

Solve one of the equations for one of its variables.

Step 2 :

Substitute the expression from step 1 into the other equation and solve for the other variable.

Step 3 :

Substitute the value from step 2 into either original equation and solve to find the value of the variable in step 1.

## Elimination Method

Step 1 :

By taking any one equations from the given two, first multiply by some suitable non-zero constant to make the co-efficient of one variable (either x or y) numerically equal.

Step 2 :

If both coefficients which are numerically equal of same sign, then we may eliminate them by subtracting those equations.

If they have different signs, then we may add both the equations and eliminate them.

Step 3 :

After eliminating one variable, we may get the value of one variable.

Step 4 :

The remaining variable is then found by substituting in any one of the given equations.

## Solving Systems of Equations by Substitution Examples

Example 1 :

Solve the following pairs of linear equations by the elimination method and the substitution method

(i) x + y = 5 and 2 x – 3 y = 4

Solution :

x + y = 5 --------(1)

2 x – 3 y = 4 --------(2)

Elimination method

(1) ⋅ 3 ==> 3x + 3y  =  15

(1) + (2)  2x - 3y  =  4

-------------------

5x  =  19

x  =  19/5

By applying x  =  19/5 in (1) equation, we get

(19/5) + y  =  5

y  =  5 - (19/5)

y  =  (25 – 19)/5

y  =  6/5

Therefore solution is x  =  19/5 and y  =  6/5

Now, let us do the same problem in substitution method :

Substitution method

x + y = 5 --------(1)

2 x – 3 y = 4 --------(2)

Step 1 :

Find the value of one variable in terms of another variable

y  =  5 – x

Step 2 :

Substitute this value of y in the other equation, and reduce it to an equation in one variable.

2x – 3(5 – x)  =  4

2x – 15 + 3 x  =  4

5x  =  4 – 15

5x  =  19

x  =  19/5

Step 3 :

Apply x  =  19/5 in the equation y = 5 – x

y  =  5 – (19/5)

y  =  (25 – 19)/5

y  =  6/5

So, the solution is x  =  19/5 and y  =  6/5

Example 2 :

Solve the following pairs of linear equations by the elimination method and the substitution method

3x + 4y  =  10 and 2x – 2y  =  2

Solution :

3x + 4y  =  10 ---------(1)

2x – 2y  =  2 ---------(2)

Elimination method

(1) + (2)    3x + 4y  =  10

(2) ⋅ 2 ==> 4 x - 4 y  =  4

----------------

7x  =  14

x  =  14/7  =  2

By applying the value of x in (2), we get

2(2) – 2y  =  2

4 – 2y  =  2

-2y  =  2 – 4

-2y  =  -2

y  =  1

Therefore solution is x  =  2 and y  =  1

Substitution method

3 x + 4 y = 10 ---------(1)

2 x – 2 y = 2 ---------(2)

Step 1 :

Find the value of one variable in terms of another variable

-2y  =  2 – 2 x

y  =  (2 x – 2)/2

y  =  x - 1

Step 2 :

Substitute this value of y in the other equation and reduce it to an equation in one variable.

3x + 4(x - 1)  =  10

3x + 4x – 4  =  10

7x – 4  =  10

7x  =  10 + 4

7x  =  14

x  =  2

Step 3 :

Apply x = 2 in the equation y = x - 1

y = 1

So, the solution is x = 2 and y = 1.

After having gone through the stuff given above, we hope that the students would have understood, solving systems of equations by elimination and substitution examples.

Apart from the stuff given in this section if you need any other stuff in math, please use our google custom search here

You can also visit our following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6