Let us recall the concepts in Pair of Linear Equation in Two Variables:
1. A pair of linear equations in two variables' can be represented, and solved, by the:
(i) graphical method
(ii) algebraic method
(i) graphical method
(ii) algebraic method
2. Graphical Method : The graph of a pair of linear equations in two variables is represented by two lines.
- When two lines in a graph intersect at only one point, then the graph is a consistent system and has one solution.
- When two lines in a graph do not intersect at any point, then the graph is an inconsistent system and has no solution.
- When two lines in a graph are identical at all points, the graph is a consistent system and has infinitely many points.
3. Algebraic Methods : We have discussed the following methods for finding the solution(s)
of a pair of linear equations :
(i) Substitution Method
(ii) Elimination Method
of a pair of linear equations :
(i) Substitution Method
(ii) Elimination Method
4. General form of the pair of linear equations in two variables:
\(a_{1}x+b_{1}y+c_{1}=0\)
\(a_{2}x+b_{2}y+c_{2}=0\)
| CONDITION | CONSISTENT OR INCOSISTENT | TYPE OF SOLUTION | GRAPHICAL REPRESENTATION |
| \(\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}\) | CONSISTENT | UNIQUE SOLUTION | INTERSECTING LINE |
| \(\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}= \frac{c_{1}}{c_{2}}\) | CONSISTENT | INFINITELY MANY SOLUTIONS | COINCIDE LINE |
| \(\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}\) | INCONSISTENT | NO SOLUTION | PARALLEL LINE |
Common mistakes to avoid:
1. Confusing the conditions for a unique solution, no solution and infinetly many solutions.
2. Errors while interpreting graphs, especially identifying intersecting, parallel and coincident lines.
3. Comparing ratios without aligning the constant term \(c\) to the same side of the equal sign in both equations [For example: using \(c_1 = -6\) for \(2x + 7y = 6\)]
4. Incorrect elimination due to multiplying only one term instead of the entire eqution.
5. Upstream and Downstream problem: Using the wrong expression for upstream speed, writing \((y-x)\) instead of \((x-y)\), where \(x\) is the speed of the boat in still water and \(y\) is the speed of the current.
6. Unit and quantity swaps in word problems: Misassigning variables in time and work or age-based problems [For example: setting "\(5\) years ago" as \(x+5\) instead of \(x-5\).
7. Errors in equations reducible to linear form: Forgetting to convert back to the original variables \(x\) and \(y\) after solving for temporary variables like \(u=\frac{1}{x}\) and \(v = \frac{1}{y}\).
Exam tips & tricks:
1. Move all terms to one side before evaluating the conditions for consistency.
2. Substitute to check: Plug your final \(x\) and \(y\) back into both equations. If \(LHS = RHS\) then the solution is correct.
3. Symmetric coefficient trick: if \(x\) and \(y\) coefficients are swapped [e.g., \(99x + 101 y = 601\) and \(101x + 99y = 599\)], add the equations once, subtract them once, then solved the simplified pair.
4. Clear fractions and deicmals first: Multiply the entire equation by the \(LCM\) of denominators (or by \(10\), \(100\)) before applying elimination or substitution.
5. The \(3\)-point graph rule: Plot \(3\) points per line, if all \(3\) don't align on a rules, you made a calculation error. Always label the intersection \((x,y)\).