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## Identify the solution(s) of the system of equations

Identify the solution(s) of the system of equations

3x+2y=4

5x-4y=3

You can use Gaussian Elimination.

Double both sides of the first equation and add the second equation.

6x + 4y = 8

5x - 4y = 3

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

11x      = 11

x = 1

5 - 4y = 3

-4y = -2

y = 1/2

There are several methods to solve systems of equations. You can solve using substitution, elimination or matrices. It depends on the nature of the equation as to which method is more efficient.

With this equation, it is most efficient to use the elimination method in the initial step. With the elimination method, the goal is to eliminate one of the variables from the equation in order to solve for the remaining variable. Once we have eliminated one of the variables, we will solve for the remaining variable.

The equations are:   3x +2y=4

5x-4y= 3

The 1st step is to get one of the variables in the equations to be inverses of each other. This will enable you to add the equations and eliminate that variable.

For the 1st equation, ( 3x+2y=4), we need to multiply both sides of the equation by 2. This will make the Y variable the inverse of the Y variable in the 2nd eqation.

When you multiply both sides of the 1st equation by 2, it becomes 6x+4y =8

Now your system of equations becomes   6x+4y=8

5x-4y=3

Now the Y variables are inverses of each other (4y, -4y).  When you add the 2 equations 4y + -4y =0. Therefore you are eliminating the Y variable.

5x-4y=3

11x +0=11

Now you can solve for x.          11x=11.

Divide both sides of the equation by 11. ( You perform the inverse function to eliminate the constant or coefficient. The inverse of multiplication is division.)

Dividing both sides by 11  provides the result   x= 1.  You have solved for the x variable.

To solve for the Y variable we will use the substitution method. The substitution method is the most efficient when you know the value of one of your variables.

The 2nd equation becomes  5(1) - 4y =3    which equates to 5 -4y =3.

We want to isolate the variable on  1 side alone.

To isolate the Y variable we will subtract 5 from both sides of the equation.

5-5-4y=3-5 (Remember whatever operation you perform on one side of the equation, you must perform the identical operation on the other side.)

The remaining result is : -4y = -2

Divide both sides of the equation by -4  (Division is the inverse operation of multiplication.)

The result is   y =  -2/-4  which is equal to 1/2

Therefore  x= 1, y = 1/2.

The last step in solving any equation is to substitute the values into the equations to ensure it is the feasible solution.

3(1) + 2(1/2) = 4

3 +1 =4     or 4 =4.

The 2nd equation:

5 (1) - 4(1/2) = 3

5- 2= 3  or 3=3.

We have checked our solution and the ordered pair (1, 1/2) satifies both equations.

Solve the first equation for y:

3x + 2y = 4

2y = 4 - 3x

y = (4 - 3x ) / 2

Substitute (4 - 3x) / 2 for y in the second equation.

5x - 4[(4 - 3x) / 2] = 3

5x - 2(4 - 3x) = 3

5x - 8  + 6x = 3

11 x = 11

x = 1

Use this value for x to solve for y using either equation.

3x + 2y = 4

3(1) + 2y = 4

3 + 2y = 4

2y = 1

y = 1/2

OR

5x - 4y = 3

5(1) -4y = 3

5 - 4y = 3

-4y = -2

y = -2/-4 = 1/2