# Approach to solve Simultaneous Equations : Substitution & Graphical Method

**Approach 2 : Substitution Method**

In this method, we express one of the variables in terms of the other in 1 of the equation and then substitute the value of the variable in the other equation. On simplifying the resulted equation, we get the value of the 1st variable. Putting the value in any of the equations and simplifying we get the vale of 2nd variable. **Note: **Always try to pick easy looking equation to express one of the variables in terms if the other.

**E.g.: 2x – 3y = 1 ……………(a) 3x – 4y = 1 ……………(b) **

First let us express one of the variables in terms of the other.

2x – 3y = 1

3y = 2x – 1

………………..(c)

Substituting this value of y in equation (b)

9x – 8x + 4 = 3

x = -1

Putting value of x in equation (c), we get

y = -1

The solution is (-1 , -1)

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Let's look at the graph of the two equations2x – 3y = 1 and 3x – 4y = 1on the same plot.

**Approach 3 : Graphical Method**

In this method, we will plot the lines represented by the two equations on the same coordinate axis. The co-ordinates of the point of intersection of the two lines will be the solution of the simultaneous equations.

Let’s learn how to plot the graph of a linear equation **2x – 3y = 1**.

Trying putting some simple values of x and get corresponding values of y.

For x = 1, y = 1/3

For x = 2, y = 1

So, (1 , 1/3) & (2 , 1) satisfies the above equation and hence is the solution of the equation 2x – 3y = 1

Now, lets put these points on the coordinate axis and draw the graph of the above equation.

Let’s plot the graph of a linear equation **3x – 4y = 1**.

Trying putting some simple values of x and get corresponding values of y.

For x = 1, y = 1/2

For x = 3, y = 2

So, (1 , 1/2) & (3 , 2) satisfies the above equation and hence is the solution of the equation 3x – 4y = 1.

Now, lets put these points on the coordinate axis and draw the graph of the above equation.

Now, let’s plot the graph of the two equations **x + 2y = 4 and 3x – y = 5** on the same co-ordinate axis to get the co-ordinate of the point of intersection to find the solution of the simultaneous equations.

So, (-1,-1) is the solution of the simultaneous linear equations **x + 2y = 4 and 3x – y = 5**.