Equations Containing Two Variables

Equations Containing Two Variables

Equations containing two variables if it is written in the form of ax + by + c=0, where a, b & c are real numbers and the coefficients of x and y, that is a and b respectively and is not equal to zero.

For example

ax+by=p 

cx+dy=q are linear equations in two variables.

The system is called linear if the variables are only to the first power and only in the numerator and there are no products of variables in any of the equations.

Example of a system with numbers.

$$\begin{array}{cc}3x-y& =7\\ 2x+3y& =1\end{array}$$

A solution to a system of equations is a value of $x$

$x$and a value of $y$

$y$that, when substituted into the equations, satisfy both equations at the same time.

For the example above $x=2$

$x=2$

 and $y=-1$

$y=-1$

 is a solution to the system. This is easy enough to check.

$$\begin{array}{cc}3\left(2\right)-\left(-1\right)& =7\end{array}$$

$$\begin{array}{cc}2\left(2\right)+3\left(-1\right)& =1\end{array}$$

 

There are two methods for solving systems of linear equations with two variables, by substitution or by elimination.

Substitution Method

In the substitution method, one equation is expressed one variable in terms of the other and then the expression is substituted in the other equation.

For example

3x + 2y = 2
y + 8 = 3x

Separate the variable y in the equation
y + 8 = 3x get y = 3x – 8.

Substitute 3x – 8 into 3x + 2y = 2.
3x + 2(3x – 8) = 2
3x + 6x – 16 = 2
9x – 16 = 2
9x = 18

Substitute x = 2 in y = 3x – 8 to get the value y
y = 3 (2) – 8
y = 6 – 8 = – 2

x = 2 and y = –2

 

 

Elimination Method

In the elimination method, the object is to make the coefficients of one variable the same in both equations so that one variable can be eliminated either by adding equations together or by subtracting one from the other.

For Example

2x + 3y = –2

4x – 3y = 14

In the above example, the coefficients of y are already opposites, and add the two equations to eliminate y.

6x = 12

Get the value of y, and substitute x = 2 into the equation 2x + 3y = –2
2(2) + 3y = –2
4 + 3y = –2
3y = –6
y = –2

 x = 2 and y = –2

 

Also, read Properties of Exponents