Pair of Linear Equations in Two Variables · Learner pages

Lesson 1 · Read two lines as two conditions

Interpret the intersection as a shared solution.

Name or initials: ____________   Date: ____________

An ordered pair (x, y) solves an equation when substitution makes the equation true.

1. Name the conditions. Use x + y = 4 and y = x. Each equation describes a straight line.
2. Plot the first line. The points (0, 4) and (4, 0) satisfy x + y = 4. Join them with a straight line.
3. Plot the second line. The points (0, 0) and (4, 4) satisfy y = x. The second line rises as x rises.
4. 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).

1. Which point lies on both x + y = 2 and y = x?

   1. (1, 1) · 2. (0, 2) · 3. (2, 0)

2. Which point lies on both x + y = 4 and y = x?

   1. (0, 4) · 2. (4, 0) · 3. (2, 2)

3. Which point lies on both x + y = 6 and y = x?

   1. (6, 0) · 2. (3, 3) · 3. (0, 6)

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.

Name or initials: ____________   Date: ____________

Adding, subtracting or dividing both sides by the same nonzero number preserves an equation.

1. Isolate a variable. For x + y = 11, write y = 11 − x.
2. Substitute into the other condition. If 2x + y = 16, replace y to get 2x + (11 − x) = 16.
3. Solve the single variable. Simplify to x + 11 = 16, hence x = 5. Then y = 11 − 5 = 6.
4. 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.

1. Solve y = x + 3 and 2x + y = 6. What is x?

   1. 1 · 2. 4 · 3. 6

2. Solve y = x + 3 and 2x + y = 9. What is x?

   1. 5 · 2. 9 · 3. 2

3. Solve y = x + 3 and 2x + y = 12. What is x?

   1. 12 · 2. 3 · 3. 6

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.

Name or initials: ____________   Date: ____________

Equal coefficients can cancel by subtraction; opposite coefficients can cancel by addition.

1. Choose a cancellation. Use x + y = 9 and x − y = 3. The y coefficients are opposites.
2. Add entire equations. Adding left sides and right sides gives 2x = 12. Thus x = 6.
3. Recover the other variable. Use 6 + y = 9 to get y = 3. Check 6 − 3 = 3.
4. 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.

1. For x + y = 4 and x − y = 2, what is y?

   1. 1 · 2. 3 · 3. 4

2. For x + y = 6 and x − y = 2, what is y?

   1. 4 · 2. 6 · 3. 2

3. For x + y = 8 and x − y = 2, what is y?

   1. 8 · 2. 3 · 3. 5

Discuss: Design one pair with a unique intersection, one parallel pair and one coincident pair. Explain each using elimination.

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. Which point lies on both x + y = 10 and y = x?

   (0, 10) / (10, 0) / (5, 5)

2. Which point lies on both x + y = 12 and y = x?

   (12, 0) / (6, 6) / (0, 12)

3. Solve y = x + 3 and 2x + y = 18. What is x?

   8 / 18 / 5

4. Solve y = x + 3 and 2x + y = 21. What is x?

   21 / 6 / 9

5. For x + y = 12 and x − y = 2, what is y?

   7 / 12 / 5

6. For x + y = 14 and x − y = 2, what is y?

   14 / 6 / 8

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. Which point lies on both x + y = 14 and y = x?

   (7, 7) / (0, 14) / (14, 0)

2. Which point lies on both x + y = 16 and y = x?

   (0, 16) / (16, 0) / (8, 8)

3. Solve y = x + 3 and 2x + y = 24. What is x?

   7 / 10 / 24

4. Solve y = x + 3 and 2x + y = 27. What is x?

   11 / 27 / 8

5. For x + y = 16 and x − y = 2, what is y?

   7 / 9 / 16

6. The second equation is exactly twice the first. How many common points do the lines have?

   Exactly two / None / Infinitely many