Pair of Linear Equations in Two Variables · Answer guide

Lesson 1 · Read two lines as two conditions

Interpret the intersection as a shared solution.

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

   Answer: (1, 1). Substitute y = x into x + y = 2: 2x = 2, so x = y = 1.

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

   Answer: (2, 2). Substitute y = x into x + y = 4: 2x = 4, so x = y = 2.

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

   Answer: (3, 3). Substitute y = x into x + y = 6: 2x = 6, so x = y = 3.

Discuss: Draw y = 1 and y = 3. Explain why two horizontal, distinct lines have no common point.

Teaching note: A point on only one line is not a solution of the pair. A rough sketch suggests a value; substitution verifies it. Open responses need discussion; they are not automatically marked.

Lesson 2 · Substitute one equation into the other

Replace one variable with an equal expression and solve.

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

   Answer: 1. Substitute y: 2x + (x + 3) = 6; 3x = 3; x = 1. Then y = 4.

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

   Answer: 2. Substitute y: 2x + (x + 3) = 9; 3x = 6; x = 2. Then y = 5.

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

   Answer: 3. Substitute y: 2x + (x + 3) = 12; 3x = 9; x = 3. Then y = 6.

Discuss: Invent two purchases using the same two item prices. Solve the resulting equations and verify the prices.

Teaching note: Use brackets when substituting an expression after a minus sign or coefficient. Open responses need discussion; they are not automatically marked.

Lesson 3 · Eliminate a variable and classify the result

Use elimination and distinguish one, no and infinitely many solutions.

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

   Answer: 1. Adding gives 2x = 6, so x = 3. Then y = 4 − 3 = 1.

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

   Answer: 2. Adding gives 2x = 8, so x = 4. Then y = 6 − 4 = 2.

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

   Answer: 3. Adding gives 2x = 10, so x = 5. Then y = 8 − 5 = 3.

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

Teaching note: Cancel complete terms, not parts of terms. An identity after elimination does not force x and y to be zero. Open responses need discussion; they are not automatically marked.

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?

   (5, 5). Substitute y = x into x + y = 10: 2x = 10, so x = y = 5.

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

   (6, 6). Substitute y = x into x + y = 12: 2x = 12, so x = y = 6.

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

   5. Substitute y: 2x + (x + 3) = 18; 3x = 15; x = 5. Then y = 8.

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

   6. Substitute y: 2x + (x + 3) = 21; 3x = 18; x = 6. Then y = 9.

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

   5. Adding gives 2x = 14, so x = 7. Then y = 12 − 7 = 5.

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

   6. Adding gives 2x = 16, so x = 8. Then y = 14 − 8 = 6.

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). Substitute y = x into x + y = 14: 2x = 14, so x = y = 7.

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

   (8, 8). Substitute y = x into x + y = 16: 2x = 16, so x = y = 8.

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

   7. Substitute y: 2x + (x + 3) = 24; 3x = 21; x = 7. Then y = 10.

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

   8. Substitute y: 2x + (x + 3) = 27; 3x = 24; x = 8. Then y = 11.

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

   7. Adding gives 2x = 18, so x = 9. Then y = 16 − 9 = 7.

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

   Infinitely many. The equations describe the same line, so each point on that line solves both.