Solve the Simultaneous Equations -3X + 4Y=11 & X-2Y = -5 to find the values of X and Y

Simultaneous Equations are equations involving 2 or more unknowns that have the same value in each equation. The aim is to eliminate all but one unknown. This will then allow us to find the value of the non eliminated unknown. From this any other unknowns can be found. There are 2 main methods we can use to solve these problems:

A) Solving by Elimination.

The first step is to label the equations

(1) -3X +4y= 11

(2)    X - 2Y= -5

You must then manipulate the equations in such a way that the coefficient of one of the unknown variables is the same value in both equations. 

In this example, the equations will be manipulated in order to obtain equal values for the coefficient of Y in both equations

Equation (2) multiplied by 2

(1):        -3X +4y= 11

(2) x 2:   2X -4Y= -10

We can now add both equations together in order to eliminate the Y unknown.

(1) + (2) x 2

(1):        -3X +4y= 11


(2) x 2:   2X -4Y= -10

-X= 1

Divide both sides by -1

X= -1

In order to find the value of Y, substitute X= -1 into any one of the original equations.

Substituting X= -1 into  equation (2):      X- 2Y = -5


-1 -2Y = -5

-2Y = -4

Divide both sides by -2


Therefore we have solved the simultaneous equations and obtained the answer of X= -1 and Y=2.

B) Solving by Substitution

The first step here is making one of the unknown variables the subject of one of the equations.

Using equation (2): X- 2Y= -5 ——> X= -5 -2Y

You must then substitute this value for the unknown into the other equation.

Substituting X= -5- 2Y into Equation (1): -3X + 4Y =11

-3 (-5+ 2Y) +4Y =11

Expand the brackets

15 -6Y + 4Y =11

Group like terms together

15-11 = 6Y-4Y

4= 2Y

Divide both sides by 2


To find the value of X, substitute  Y=2 into X= -5- 2Y

X= -5 - 2(2)

X= -1

Therefore we have solved the simultaneous equations and obtained the answer of x= -1 and y=2.

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