Simultaneous linear equations up to three variables

When two or three variables are together in two linear equations, it is a simultaneous linear equation. It is in the ordered pair (x, y). We take these equations to find a standard solution. For example, we draw two linear graphs and find out their intersection point.

For finding values of two or more equations, we need some algebraic skills. We call these equations a simultaneous equation because we solve them simultaneously.

All we do is eliminate or remove one of the variables to solve a single equation. Here are a few steps following which you can solve a simultaneous equation.

• We got two variables. To get rid of one, we use the elimination method.
• Then, we find the value of the left variable.
• Then substitute the value of the remaining part.
• We get the final answer.
•  Now place the answer in the given equations to check the accuracy.

We can understand this by solving some equations,

1.     3x + 2y = 11

2x + 3y = 4, find the value of x and y?

Solution:

The given equations are

3x + 2y = 11 (i)

2x + 3y = 4 (ii)

From (i) 2y=11−3x

⇒y=211−3x​ ……….(iii)

Substituting the worth of y from (iii) in (ii),

we get,

2x+3(211−3x​) = 4

⇒4x+33−9x=8

⇒−5x=−25

⇒x=5

Putting x=5 in (iii), we get 

y=211−3×5​=211−15​

=2−4​=−2

Hence the required solution is x=5 and y=−2.

There are two types of elementary examples.

• Trivial example

This involves one equation, and the other one is unknown.

4x = 8

Solution will be, x = 2

• Non-trivial example

This includes two variables and two equations.

2x + 3y = 6

4x + 9y = 15

We can solve this by keeping x in terms of y.

Therefore, x = 3 – 3/2y.

Now placing the value of x in the bottom equation,

4(3-3/2y) + 9y = 15

This gives y =1, now substituting the value of y in the previous equation.

This gives, x = 3/2.

Simultaneous Linear Equation in Two Variables

When the equation is presented in the form of Ax+By+C = 0, it is a linear equation in two variables. Here A, B, and C represent real numbers, and y and x are variables. A and B are non-zero numbers.

Generally, in mathematics, these equations are to find the coordinates in geometry for a straight line.

Let us consider an equation, x + y – 3 = 0

When x = 0, y = 3

When x = 1, y=2

When x = 2, y= 1

When x = 3, y= 0

When x = 7, y= -4

For the given equation, any of these can be the solution.

Different Methods of Solving Linear Equation in Two Variables

There are many different methods by which we can solve simultaneous linear equations in two variables.

• Elimination method

In this, we eliminate any one variable. This method is easier than the substitution method.

1-Solve the system of equation 2x + 3y = 11, x + 2y = 7 by the method of elimination.

Solution:

The given equations are:

2x + 3y = 11       …………… (i)

x + 2y = 7       …………… (ii)

Multiply the equation (ii) by 2, we get

x + 2y = 7   …………… (× 2)

2x + 4y = 14       …………… (iii)

Subtract equations (i) and (ii), we get

-y = -3

Substituting the worth of y = 3 in (i), we get 

2x + 3y = 11 

2x + 3 × 3 = 11

2x + 9 = 11

2x + 9 – 9 = 11 – 9

2x = 2 

 x = 1

Therefore, the system of the given equations are x = 1and y = 3.

• Substitution method

When we get two equations, then by this method, we can find the value of one in terms of another. This is the first step. Let us assume two simultaneous linear equations,

x + y=7, x-y=8

then, x=7-y

• Determinant method

We solve in the form of dy and dx. In the form of a matrix.

For a 2×2 2 × 2 matrices, [abcd] [ a b c d ], the determinant ∣∣∣abcd∣∣∣ | a b c d | is to be ad−bc a d − bc.

• Cross-multiplication method

In this, we cross multiply the fractions—the numerator of one side with the denominator of the other side and vice versa.

1- 8x + 5y = 11

3x – 4y = 10

Solution:

By cross-multiplication method:

x/(5) (-10) – (-4) (-11) = y/(-11) (3) – (-10) (8) = 1/(8) (-4) – (3) (5)

x/-50 – 44 = y/-33 + 80 = 1/-32 – 15

x/-94 = y/47 = 1/-47

x/-2 = y/1 = 1/-1 [ multiplying by 47 ]

x = -2/-1 = 2 and y = 1/-1 = -1

Therefore, the solution is x = 2, y = -1

Simultaneous Linear Equation in Three Variables

Understand this as a triplet of the equation containing multiple variables. We express this in ax + by + cz = d., also called an ordered triple. This is a simultaneous linear equation in three variables.

a, b, c = non-zero coefficients.

x, y, z = unknown variables.

D being the constant.

To solve these equations, we need to follow a few steps. Firstly, achieve back-substitution by eliminating one variable at a time.

A simple example for linear equations in three variables.

• 3x+2y−z=6−2x+2y+z=3x+y+z=4

Solution  

 x=1 y=2 z=1

There are different methods of solving simultaneous linear equations in three variables. These are the elimination method, substitution method, graphical method, etc.

Conclusion

This is all about simultaneous linear equations in two and three variables.

After solving more and more questions related to linear equations you will be pro at it. It is an equation for the straight line. We hope now maths is a fun task for you.