i. x - 2y = 4
∴ 2y = x - 4
∴ y = (x - 4)/2
| x |
0 |
2 |
-2 |
4 |
| y |
-2 |
-1 |
-3 |
0 |
| (x, y) |
(0, -2) |
(2, -1) |
(-2, -3) |
(4, 0) |
2x - 4y = 12
∴ x -2y = 6
∴ 2y = x - 6
∴ y = (x - 6)/2
| x |
0 |
-2 |
2 |
4 |
| y |
-3 |
-4 |
-2 |
-1 |
| (x, y) |
(0, -3) |
(-2, -4) |
(2, -2) |
(4, -1) |
ii. Ratio of coefficients of x = 1/2
Ratio of coefficients of y \(=\frac{-2}{-4} = \frac{1}{2}\)
Ratio of constant terms \(=\frac{4}{12} = \frac{1}{3}\)
∴ Ratio of coefficients of x = ratio of coefficients of y ratio of constant terms
iii. If ratio of coefficients of x = ratio of coefficients of y ≠ ratio of constant terms, then the graphs of the two equations will be parallel to each other.
Condition of consistency in Equations:
| Sr. No. |
Simultaneous Equations |
\(\frac{a_1}{a_2}\) |
\(\frac{b_1}{b_2}\) |
\(\frac{c_1}{c_2}\) |
Comparison of ratios |
Graphical Interpretation |
Algebraic Interpretation |
| 1. |
x + y = 3; x - y = 1 |
\(\frac{1}{1}\) |
\(\frac{1}{-1}\) |
\(\frac{3}{1}\) |
\(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\) |
Intersecting lines |
Unique solution (OR) Only one common solution |
| 2. |
2x - y = -1; 2x - y = 4 |
\(\frac{2}{2}\) |
\(\frac{-1}{-1}\) |
\(\frac{-1}{4}\) |
\(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\) |
Parallel lines |
No solution |
| 3. |
x - y = -2; 2x - 2y = -4 |
\(\frac{1}{2}\) |
\(\frac{-1}{-2}\) |
\(\frac{-2}{-4}\) |
\(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) |
Coincident lines |
Infinity many solutions |
