Introduction
This chapter deals with pairs of linear equations in two variables. You will learn various methods to solve such pairs of equations: graphical method, substitution method, and elimination method. You will also learn to interpret the solutions geometrically and understand when a pair of equations has a unique solution, infinitely many solutions, or no solution.
Graphical Representation and Consistency
A pair of linear equations in two variables can be represented graphically as two straight lines. The nature of the solution depends on how these lines relate to each other. If the lines intersect at a point, there is exactly one solution (consistent pair). If the lines are coincident (overlap completely), there are infinitely many solutions (dependent pair). If the lines are parallel, there is no solution (inconsistent pair).
Key Points
- •Intersecting lines: unique solution (consistent)
- •Coincident lines: infinitely many solutions (dependent and consistent)
- •Parallel lines: no solution (inconsistent)
- •For a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, compare a1/a2, b1/b2, c1/c2
- •Unique solution: a1/a2 is not equal to b1/b2
Worked Example
Check consistency: 2x + 3y = 5 and 4x + 6y = 10. a1/a2 = 2/4 = 1/2 b1/b2 = 3/6 = 1/2 c1/c2 = 5/10 = 1/2 Since a1/a2 = b1/b2 = c1/c2, the lines are coincident. Infinitely many solutions.
Watch Out
Remember the three conditions: a1/a2 != b1/b2 (unique), a1/a2 = b1/b2 = c1/c2 (infinite), a1/a2 = b1/b2 != c1/c2 (no solution).
Substitution Method
In the substitution method, we express one variable in terms of the other from one equation and substitute this expression in the other equation. This reduces the system to a single equation in one variable, which can be solved easily. Then we substitute back to find the other variable.
Key Points
- •Step 1: Express one variable in terms of the other from one equation
- •Step 2: Substitute into the other equation to get a single-variable equation
- •Step 3: Solve for that variable
- •Step 4: Substitute back to find the other variable
- •Choose the equation and variable that makes the algebra simplest
Worked Example
Solve: x + 2y = 8 ...(1) and 2x - 3y = 2 ...(2) From (1): x = 8 - 2y Substitute in (2): 2(8 - 2y) - 3y = 2 16 - 4y - 3y = 2 -7y = -14 y = 2 Substitute back: x = 8 - 2(2) = 4 Solution: x = 4, y = 2
Elimination Method
In the elimination method, we multiply the equations by suitable numbers so that the coefficients of one of the variables become equal (or negatives of each other). We then add or subtract the equations to eliminate that variable, leaving a single equation in one variable. This is often faster than substitution when coefficients are already close in value.
Key Points
- •Multiply equations to make coefficients of one variable equal
- •Add or subtract equations to eliminate that variable
- •Solve the resulting single-variable equation
- •Substitute back to find the other variable
- •This method is preferred when substitution leads to fractions
Worked Example
Solve: 3x + 4y = 10 ...(1) and 2x - 2y = 2 ...(2) Multiply (2) by 2: 4x - 4y = 4 ...(3) Add (1) and (3): 7x = 14, so x = 2 Substitute in (1): 6 + 4y = 10, so y = 1 Solution: x = 2, y = 1
Watch Out
When coefficients of one variable are already equal or close, elimination is faster than substitution.
Word Problems on Linear Equations
Many real-life situations can be modelled using a pair of linear equations. The key skill is translating the word problem into mathematical equations. Common types include age problems, number problems, speed-distance-time problems, and fraction problems. Always define your variables clearly and write two independent equations.
Key Points
- •Define variables clearly at the start
- •Identify two independent conditions from the problem
- •Form two equations from these conditions
- •Solve using substitution or elimination
- •Verify the solution makes sense in the context of the problem
Worked Example
The sum of two numbers is 75. If one exceeds the other by 15, find the numbers. Let the numbers be x and y (x > y). Equation 1: x + y = 75 Equation 2: x - y = 15 Adding: 2x = 90, so x = 45 Substituting: y = 75 - 45 = 30 The numbers are 45 and 30.
Quick Summary
- ✓A pair of linear equations can have: unique solution, infinitely many solutions, or no solution
- ✓Unique solution: a1/a2 != b1/b2 (intersecting lines)
- ✓Infinitely many: a1/a2 = b1/b2 = c1/c2 (coincident lines)
- ✓No solution: a1/a2 = b1/b2 != c1/c2 (parallel lines)
- ✓Substitution: express one variable, substitute in other equation
- ✓Elimination: make coefficients equal, add/subtract to eliminate one variable
- ✓Always verify your solution by substituting in both original equations
Key Formulas
Consistency check: compare a1/a2, b1/b2, c1/c2
Unique solution: a1/a2 != b1/b2
Infinite solutions: a1/a2 = b1/b2 = c1/c2
No solution: a1/a2 = b1/b2 != c1/c2
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