# SOLVING SYSTEM OF EQUATIONS

We can use one of the following methods to solve a system of linear equations.

1. Elimination Method

2. Substitution Method

3. Cross Multiplication Method

4. Graphing

## Solved Examples

Example 1 :

Solve by elimination method.

3x + 4y  =  7

x - 4y  =  -3

Solution :

3x + 4y  =  7 -----(1)

x - 4y  =  -3  -----(2)

In the given two equations, y-term has the same coefficient and different signs. By adding the above two equations, we can eliminate y-term and solve for y.

(1) + (2) :

4x  =  4

Divide each side by 4.

x  =  1

Substitute x  =  1 in (1).

3(1) + 4y  =  7

3 + 4y  =  7

Subtract 3 from each side.

4y  =  4

Divide each side by 4.

y  =  1

So, the solution is

(x, y)  =  (1, 1)

Example 2 :

Solve by elimination method.

3x + 4y  =  -25

2x - 3y  =  6

Solution :

3x + 4y  =  -25 -----(1)

2x - 3y  =  6  -----(2)

Both x terms and y terms have different coefficients in the above system of equations.

Let's try to make the coefficients of y terms equal.

To make the coefficients of y terms equal, we have to find the least common multiple 4 and 3.

The least common multiple of 4 and 3 is 12.

Multiply the first equation by 3 in order to make the coefficient of y as 12 and multiply the second equation by 4 in order to make the coefficient of y as -12.

(1) ⋅ 3 ---->  9x + 12y  =  -75

(2)  4 ---->  8x - 12y  =  24

Now, we can add the two equations and eliminate y as shown below. Divide each side by 17.

x  =  -3

Substitute -3 for x in (1).

(1)---->  3(-3) + 4y  =  -25

-9 + 4y  =  -25

4y  =  -16

Divide each side by 4.

y  =  -4

So, the solution is

(x, y)  =  (-3, -4)

Example 3 :

Solve for x and y using substitution.

x - 5y + 17  =  0

2x + y + 1  =  0

Solution :

x - 5y + 17  =  0 -----(1)

2x + y + 1  =  0 -----(2)

Step 1 :

Solve (1) for x.

x - 5y + 17  =  0

Subtract 17 from each side.

x - 5y  =  -17

x  =  5y - 17 -----(3)

Step 2 :

Substitute (5y - 17) for x into (2).

(2)-----> 2(5y - 17) + y + 1  =  0

10y - 34 + y + 1  =  0

11y - 33  =  0

11y  =  33

Divide each side by 11.

y  =  3

Step 3 :

Substitute 3 for y into (3).

(3)-----> x  =  5(3) - 17

x  =  15 - 17

x  =  -2

So, the solution is

(x, y)  =  (-2, 3)

Example 4 :

Solve the following system of equations using cross multiplication method.

2x + 7y - 5  =  0

-3x + 8y  =  -11

Solution:

First we have to change the given linear equations in the form a1x + b1y + c1  =  0, a2x + b2y + c2  =  0.

2x + 7y - 5  =  0

-3x + 8y + 11  =  0 x/(77 + 40)  =  y/(15 - 22)  =  1/[16 + 21]

x/117  =  y/(-7)  =  1/37

 x/117  =  1/37x  =  117/37 y/(-7)  =  1/37y  =  -7/37

So, the solution is

(x, y)  =  (117/37, -7/37)

Example 5 :

Solve the following system of equations by graphing.

x + y - 4  =  0

3x - y  =  0

Solution :

Step 1 :

Let us re-write the given equations in slope-intercept form  (y = mx + b).

y  =  - x + 4

(slope is -1 and y-intercept is 4)

y  =  3x

(slope is 3 and y-intercept is 0)

Based on slope and y-intercept, we can graph the given equations. Step 2 :

Find the point of intersection of the two lines. It appears to be (1, 3). Substitute to check if it is a solution of both equations.

 x + y - 4  =  0 1 + 3 - 4  =  0  ?4 - 4  =  0  ?0  =  0  True 3x - y  =  0 3(1) - 3  =  0  ?3 - 3  =  0  ?0  =  0  True

Because the point (1, 3) satisfies both the equations, the solution for the given system is (1, 3). Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

v4formath@gmail.com

You can also visit the 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

1. Click on the HTML link code below.

Featured Categories

Math Word Problems

SAT Math Worksheet

P-SAT Preparation

Math Calculators

Quantitative Aptitude

Transformations

Algebraic Identities

Trig. Identities

SOHCAHTOA

Multiplication Tricks

PEMDAS Rule

Types of Angles

Aptitude Test 