Pair of Linear Equations in Two Variables · Lesson 4 of 4
Chapter Summary and Practice
“Graphs and algebra join forces to settle pairs of equations from every angle.”
• Recall the meaning and general form of a linear equation in two variables. • Interpret the solution of a pair of linear equations graphically and algebraically. • Distinguish between intersecting, parallel and coincident lines. • Choose an appropriate method to solve a pair of linear equations. • Apply substitution and elimination methods accurately.
In this chapter, we studied pairs of linear equations in two variables and learned how to interpret and solve them using graphical and algebraic methods. This final lesson brings together the main ideas, formulas, solution cases and problem-solving strategies.
1. Linear Equations in Two Variables
A linear equation in two variables is an equation that can be written in the standard form shown below, where a and b are not both zero.
An ordered pair (x, y) is a solution if substituting its values into the equation makes the equation true.
2. Pair of Linear Equations
A pair of linear equations consists of two equations involving the same variables. Their common solution must satisfy both equations at the same time.
3. Graphical Interpretation
Each linear equation in two variables represents a straight line. The solution of a pair is determined by the relationship between the two lines.
The lines meet at exactly one point. The pair has one unique solution and is consistent.
The lines never meet. The pair has no solution and is inconsistent.
Both equations represent the same line. The pair has infinitely many solutions and is dependent and consistent.
4. Substitution Method
Express one variable in terms of the other variable.
Substitute the expression into the second equation and solve the resulting single-variable equation.
Substitute the obtained value back to find the remaining variable.
Use substitution when one variable already has coefficient 1 or −1, or when one variable can be isolated without introducing difficult fractions.
5. Elimination Method
Multiply one or both equations so that the coefficients of one variable become equal in magnitude.
Add or subtract the equations to eliminate that variable.
Solve for the remaining variable and substitute back to find the other variable.
Use elimination when the coefficients are already equal, opposite or can be made equal using small multipliers.
6. Special Algebraic Results
A false numerical statement means the equations have no common solution. The corresponding lines are parallel.
A true numerical statement means the equations represent the same line and have infinitely many solutions.
7. Choosing the Right Method
| Situation | Recommended Method |
|---|---|
| One variable has coefficient 1 or −1 | Substitution |
| Coefficients are already equal or opposite | Elimination |
| You need a visual interpretation | Graphical method |
| The exact answer contains fractions or decimals | Substitution or elimination |
| The question asks about the number of solutions only | Compare coefficient ratios |
8. Common Mistakes to Avoid
When subtracting equations, change the sign of every term in the equation being subtracted.
When multiplying an equation, multiply every term on both sides by the chosen constant.
Use parentheses when replacing a variable with an expression, especially if the expression contains subtraction.
Substitute the final values into both original equations whenever possible.
Guided Practice
Problem
Solve x + y = 7 and x − y = 1.
- 1.From x + y = 7, write x = 7 − y.
- 2.Substitute into x − y = 1.
- 3.Solve 7 − y − y = 1 to get y = 3.
- 4.Substitute y = 3 into x = 7 − y to get x = 4.
Problem
Solve 2x + 3y = 12 and 2x − y = 4.
- 1.Subtract the second equation from the first.
- 2.The x-terms cancel and 4y = 8.
- 3.Therefore, y = 2.
- 4.Substitute y = 2 into 2x − y = 4 to obtain x = 3.
For every question, first identify whether you need to solve the equations, determine the number of solutions, form equations from a word problem or verify a given ordered pair.
Quiz
Which condition shows that a pair of linear equations has exactly one solution?
What does the result 0 = 12 mean while solving a pair of linear equations?
Which method is usually most convenient when one variable has coefficient 1 or −1?
Solve the equations x + y = 11 and x − y = 3.
If two equations represent the same straight line, how many common solutions do they have?
Chapter Practice
Attempt each question before reading the answer and explanation. Use substitution, elimination, graphical interpretation or coefficient comparison as appropriate.
Problem
Which ordered pair satisfies the equation 2x + y = 7?
A. (1, 3)
B. (2, 3)
C. (3, 3)
D. (0, 5)
- 1.Answer: (2, 3)
- 2.Explanation: Substituting x = 2 and y = 3 gives 2(2) + 3 = 7.
Problem
If two lines intersect at exactly one point, the pair of equations has:
A. No solution
B. One unique solution
C. Infinitely many solutions
D. Only integer solutions
- 1.Answer: One unique solution
- 2.Explanation: The point of intersection satisfies both equations.
Problem
Which condition represents parallel lines?
A. a₁/a₂ ≠ b₁/b₂
B. a₁/a₂ = b₁/b₂ = c₁/c₂
C. a₁/a₂ = b₁/b₂ ≠ c₁/c₂
D. a₁ = a₂ only
- 1.Answer: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
- 2.Explanation: Equal x- and y-coefficient ratios but a different constant ratio represents distinct parallel lines.
Problem
Solve x + y = 9 and x − y = 3.
- 1.Answer: x = 6, y = 3
- 2.Explanation: Adding the equations gives 2x = 12, so x = 6. Then y = 3.
Problem
Solve 3x + 2y = 16 and x + y = 6 using substitution.
- 1.Answer: x = 4, y = 2
- 2.Explanation: From x + y = 6, x = 6 − y. Substitution gives 18 − 3y + 2y = 16, so y = 2 and x = 4.
Problem
Solve 2x + 3y = 13 and 4x − 3y = 5 using elimination.
- 1.Answer: x = 3, y = 7/3
- 2.Explanation: Adding the equations gives 6x = 18, so x = 3. Substitution gives 6 + 3y = 13, so y = 7/3.
Problem
Determine the number of solutions of 2x + 4y = 8 and x + 2y = 4.
- 1.Answer: Infinitely many solutions
- 2.Explanation: The first equation is exactly twice the second equation.
Problem
Determine the number of solutions of 3x + 6y = 9 and x + 2y = 5.
- 1.Answer: No solution
- 2.Explanation: The coefficients are proportional, but the constants are not proportional, so the lines are parallel.
Problem
Check whether (2, 1) is a common solution of x + y = 3 and 2x − y = 3.
- 1.Answer: Yes
- 2.Explanation: The pair satisfies both equations: 2 + 1 = 3 and 2(2) − 1 = 3.
Problem
The sum of two numbers is 25 and their difference is 7. Find the numbers.
- 1.Answer: 16 and 9
- 2.Explanation: Let the numbers be x and y. Solve x + y = 25 and x − y = 7.
Problem
Two notebooks and three pens cost ₹84. One notebook and two pens cost ₹49. Find the cost of each item.
- 1.Answer: Notebook = ₹21, Pen = ₹14
- 2.Explanation: Let notebook cost be x and pen cost be y. Solve 2x + 3y = 84 and x + 2y = 49.
Problem
The sum of the ages of a father and son is 50 years. The father is four times as old as the son. Find their ages.
- 1.Answer: Father = 40 years, Son = 10 years
- 2.Explanation: Let their ages be x and y. Solve x + y = 50 and x = 4y.
Problem
While eliminating variables, you obtain 0 = 12. What does this tell you about the pair of equations?
- 1.Answer: The pair has no solution.
- 2.Explanation: The false statement indicates distinct parallel lines.
Problem
While solving a pair, you obtain 0 = 0. What additional conclusion can you make?
- 1.Answer: The equations represent the same line and have infinitely many solutions.
- 2.Explanation: The system is dependent and consistent.
Problem
For what value of k will the equations 2x + 3y = 7 and 4x + 6y = k have infinitely many solutions?
- 1.Answer: k = 14
- 2.Explanation: The second equation must be exactly twice the first equation.
Problem
For what value of k will the equations x + 2y = 5 and 2x + 4y = k have no solution?
- 1.Answer: Any value of k except 10
- 2.Explanation: The coefficient ratios are equal. The lines are coincident only when k = 10; otherwise they are parallel.
Frequently Asked Questions
Yes, substitution can be used in principle, but elimination may produce simpler calculations in many cases.
Add the equations when the target coefficients are opposites. Subtract when the target coefficients are equal and have the same sign.
A solution of the pair must be common to both equations, not just one of them.
No. It means the two equations are dependent and represent the same line, so infinitely many ordered pairs satisfy both.
Key Takeaways
• Always write the equations in a clear standard form before comparing their coefficients. • Choose the solution method that produces the simplest arithmetic. • Use parentheses carefully when substituting an expression for a variable. • Multiply every term when modifying an equation. • Check the final solution in both original equations. • Interpret complete cancellation instead of treating it as an error.
Previous · Lesson 3
Algebraic Method 2: The Elimination Method
Next
End of chapter