Three guided lessons with explanations, visual steps, practice and a final check.
Original learning companion, not full textbook exercise solutions. Covers graph interpretation, substitution, elimination and consistency through selected original problems. It is not an exhaustive word-problem or textbook solution bank.
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LESSON 1
Read two lines as two conditions
Interpret the intersection as a shared solution.
An ordered pair (x, y) solves an equation when substitution makes the equation true.
Name the conditions
Use x + y = 4 and y = x. Each equation describes a straight line.
Plot the first line
The points (0, 4) and (4, 0) satisfy x + y = 4. Join them with a straight line.
Plot the second line
The points (0, 0) and (4, 4) satisfy y = x. The second line rises as x rises.
Read and verify
The lines meet at (2, 2). Check 2 + 2 = 4 and 2 = 2; both are true.
Worked example: For x + y = 8 and y = x, substitute to get 2x = 8. Thus x = 4, y = 4 and the intersection is (4, 4).
Watch for this: A point on only one line is not a solution of the pair. A rough sketch suggests a value; substitution verifies it.
Discuss: Draw y = 1 and y = 3. Explain why two horizontal, distinct lines have no common point.
LESSON 2
Substitute one equation into the other
Replace one variable with an equal expression and solve.
Adding, subtracting or dividing both sides by the same nonzero number preserves an equation.
Isolate a variable
For x + y = 11, write y = 11 − x.
Substitute into the other condition
If 2x + y = 16, replace y to get 2x + (11 − x) = 16.
Solve the single variable
Simplify to x + 11 = 16, hence x = 5. Then y = 11 − 5 = 6.
Check both conditions
5 + 6 = 11 and 2 × 5 + 6 = 16. The ordered pair is (5, 6).
Worked example: Two notebooks and one pen cost 34 units; one notebook and one pen cost 22 units. Let costs be n and p. From p = 22 − n, 2n + 22 − n = 34, so n = 12 and p = 10. Both purchases check.
Watch for this: Use brackets when substituting an expression after a minus sign or coefficient.
Discuss: Invent two purchases using the same two item prices. Solve the resulting equations and verify the prices.
LESSON 3
Eliminate a variable and classify the result
Use elimination and distinguish one, no and infinitely many solutions.
Equal coefficients can cancel by subtraction; opposite coefficients can cancel by addition.
Choose a cancellation
Use x + y = 9 and x − y = 3. The y coefficients are opposites.
Add entire equations
Adding left sides and right sides gives 2x = 12. Thus x = 6.
Recover the other variable
Use 6 + y = 9 to get y = 3. Check 6 − 3 = 3.
Interpret exceptional results
A false statement such as 0 = 4 means no solution. An identity such as 0 = 0 after reducing equivalent equations means the same line and infinitely many solutions.
Worked example: For 2x + y = 7 and 4x + 2y = 15, doubling the first equation gives 4x + 2y = 14. Subtracting the second gives 0 = −1, so the pair has no solution.
Watch for this: Cancel complete terms, not parts of terms. An identity after elimination does not force x and y to be zero.
Discuss: Design one pair with a unique intersection, one parallel pair and one coincident pair. Explain each using elimination.
Original learning companion, not full textbook exercise solutions. Covers graph interpretation, substitution, elimination and consistency through selected original problems. It is not an exhaustive word-problem or textbook solution bank.
Original HelioLearn teaching and questions. Source-aligned companion; not an official NCERT/CBSE publication or full textbook solution bank. Independent teacher review and narrated video are pending.
