Linear Equation in Two Variables

An equation of the form $ax + by + c = 0$, where $a, \ b, \ and \ c$ are real numbers $(a \ne 0, \ b \ne 0)$ is called a linear equation in two variables $x \ and \ y$. Such an equation has degree 1.

Examples: $2x + 5y = 10$ $-2x + 3y - 5 = 0$ $78x + (\frac{1}{2})y - 23 = 0$

It is not necessary to represent the variables by $\ x \ and \ y$ only. You could use other notations as well as $\ a \ and \ b \ or\ m \ and \ n$ or others. In our discussion we will use $\ x \ and \ y$ as these are the most common way of representing variables.

Simultaneous Linear Equations in Two Variables

Two linear equations in two variables $\ x \ and \ y$  are said to form a system of simultaneous linear equations if each of them is satisfied by the same pair of values of $\ x \ and \ y$ .

A solution to a linear system is an assignment of numbers to the variables $x \ and \ y$ such that all the equations are simultaneously satisfied.

Examples: $5x + 3y = 12 \ and \ 3x + 5y = 15$

or $4x - 5y = 20 \ and \ 6x + 5y = -2$

METHODS OF SOLVING SIMULTANEOUS LINEAR EQUATIONS

Substitution Method

Suppose we are given two linear equations in $\ x \ and \ y$. Then, we may solve them by Substitution Method as explained below:

1. Express $\ y \$ in terms of $x$ from one of the given equations.
2. Substitute this value of $\ y \$    in the second equation to obtain an equation in $\ x \$. Solve it for $\ x \$      .
3. Substitute the value of $\ x \$ in the relation obtained in step 1, to get the value of $\ y \$   .

Note: We may interchange the roles of $\ x\$ and $\ y \$    in the above method.

Example showing substitution Method:

Let the two equations be: $2x + y = 10 \ \ \ i)$ $x + 2y = 8 \ \ \ ii)$

From i) we get $y = 10 - 2x$,

Substitute this in ii) $x + 2(10 - 2x) = 8$ $20 - 3x = 8$ $3x = 12 or x = 4$

Now substituting $x=4$ in the expression for $y$ $y = 10 - 2(4) = 2$

Hence $x=4$ and $y=2$ is the solution for the two equations.

Solution also means that these lines intersect at that point.

Elimination Method

1. Multiply the given equations by suitable constants so as to make the coefficients of one of the variables, numerically equal.
2. Add the new equations, if the numerically equal coefficients are opposite in sign; otherwise subtract them.
3. Solve the equation so obtained. This gives the value of one of the variables.
4. Substitute the value of this unknown in any of the given equations. Solve it to get the value of the other variable.

Example showing substitution Method:

We will solve the same equations

Let the two equations be: $2x + y = 10 \ \ \ i)$ $x + 2y = 8 \ \ \ ii)$

Multiply the second equation by 2. We get $2x + 4y = 16 \ \ \ iii)$

Now subtract iii) from i) $2x + y = 10$ $-(2x + 4y = 16)$

We get $-3y= -6$

or $y=2$

Substituting in i), we get $x=4$

Hence $x=4$ and $y=2$ is the solution for the two equations.

Solution also means that these lines intersect at that point.