Linear Equations
For any linear equation, that is, each solution ${(x, y)}$ of a linear equation in two variables, ${ax + by + c = 0}$, corresponds to a point on the line representing the equation, and vice versa.
The general form for a pair of linear equations in two variables x and y is
${a_1x + b_1 y + c_1 = 0}$ and
${a_2x + b_2 y + c_2 = 0}$,
Some examples of pair of linear equations in two variables are:
${2x + 3y – 7 = 0}$ and ${9x – 2y + 8 = 0}$
${5x = y }$ and ${–7x + 2y + 3 = 0}$
${x + y = 7 }$ and ${17 = y}$
${2x + 3y = 9 }$ (1)
${4x + 6y = 18 }$ (2)
To obtain the equivalent geometric representation, we find two points on the line
representing each equation. That is, we find two solutions of each equation.
We plot these points in a graph paper and draw the lines. We find that both the lines coincide (see Fig. 3.3). This is so, because, both the equations are equivalent, i.e., one can be derived from the other.
Example : Two rails are represented by the equations ${x + 2y – 4 = 0}$ and ${2x + 4y – 12 = 0}$. Represent this situation geometrically.
| $x$ | 0 | 4.5 |
| $y=(9 − 2x)/3$ | 3 | 0 |
| ${x}$ | 0 | 3 |
| ${y=(18-4x)/6}$ | 3 | 1 |
We plot these points in a graph paper and draw the lines. We find that both the lines coincide (see Fig. 3.3). This is so, because, both the equations are equivalent, i.e., one can be derived from the other.
Example : Two rails are represented by the equations ${x + 2y – 4 = 0}$ and ${2x + 4y – 12 = 0}$. Represent this situation geometrically.