# 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.

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