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Math lesson plan · 9th grade (Year 10 in England) · 60 minutes

Solve a linear system by substitution

Algebra

CCSS HSA.REI.C.6England: GCSE Foundation AlgebraEngland: GCSE Higher Algebra

Objective and success criteria

Students will: Solve a linear system by substitution.

  • I can answer questions on today's skill on my own, without help.
  • I can explain each step and say why it works.
  • I can check my answer and spot a mistake.

Before and after

Needs first: Systems of linear equations (simultaneous equations)

Leads to: the next skill in this area.

Materials: Balance scales or a balance drawing, Algebra tiles, Function machine cards, Mini whiteboards and pens, Printed student sheet (below).

Warm-up · 6 min

  • Quick retrieval from earlier learning (Systems of linear equations (simultaneous equations)). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. Solve the system: y = −2x − 3 and x + 3y = −4. Find x. → x = -1Prompt: Substitute the first equation into the second.
  2. Solve: 2x + 3y = 5 and 2x − 3y = −1. Find y. → 1Prompt: The y terms have opposite coefficients: add the equations to eliminate y.
  3. Solve the system: y = −3x + 2 and 3x + 2y = −11. Find x. → x = 5Prompt: Substitute the first equation into the second.

Hook · 6 min

  • Show a "think of a number" puzzle and ask the class to work out the starting number.
  • Share the objective: Solve a linear system by substitution. Read the success criteria together.

I do: model it · 12 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: Solve the system: y = −3x − 16 4x − 3y = −17 Type the solution like (2, −1).
  • Step 1: 4x − 3(−3x − 16) = −17
  • Step 2: 13x = −65 → x = −5
  • Step 3: y = −1
  • Step 4: (−5, −1)
  • Answer: (−5, −1)
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 12 min

  • Do these together. Students try each step on whiteboards before you reveal it.
  • Use the hints as prompts rather than giving the answer.
  1. Solve the system: y = −x + 9 −3x − 2y = −21 Type the solution like (2, −1). → (3, 6)Prompt: Replace y in the second equation with the first expression.
  2. Solve the system: x = −3y + 3 x − 5y = −13 Type the solution like (2, −1). → (−3, 2)Prompt: x is already on its own: substitute it into the other equation.

You do: independent practice · 17 min

  • Students work on the student sheet. Circulate and check the first two answers of every student early.
  • Students who finish: write a question of their own for a partner, with the answer.

The 8 questions are on the student sheet below, with answers in the key.

Exit ticket · 7 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. To solve y = 3x − 1 and x + 5y = 9 by substitution, which is the correct next line? (x + 5·3x − 1 = 9 / 3x − 1 + 5y = 9 / x + 5(3x − 1) = 9 / x − 5(3x − 1) = 9) → x + 5(3x − 1) = 9Prompt: Substitution replaces y with the whole expression it equals.
  2. Solve the system: y = x −5x − 4y = 9 Type the solution like (2, −1). → (−1, −1)Prompt: Replace y in the second equation with the first expression.

Watch out for

  • Doing an operation to one side of an equation but not the other.
  • Reading 3x as 3 + x, or thinking x always stands for one particular number.

Support and stretch

Support

  • Review first: Systems of linear equations (simultaneous equations).
  • Use concrete materials: balance scales or a balance drawing.
  • Give the first step of each question, or a partly worked example to finish.
  • Pair with a confident partner for the guided practice.

Stretch

  • Explain a common mistake in writing and how to fix it.
  • Write a word problem that uses today's skill, then solve a partner's.

Homework and more practice

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-9.

Free lesson plan from eurekami.com/lesson-plan-builder

eurekami

Solve a linear system by substitution

Name

Date

Score/ 8

  1. Question 1

    To solve y = x − 1 and 2x + 5y = −6 by substitution, which is the correct next line?

    • A. 2x − 5(x − 1) = −6
    • B. 2x + 5·x − 1 = −6
    • C. x − 1 + 5y = −6
    • D. 2x + 5(x − 1) = −6

  2. Question 2

    Solve the system: y = −2x − 12 −4x − y = 20 Type the solution like (2, −1).

    Answer

  3. Question 3

    Solve the system: x = −2y − 1 −3x + y = −4 Type the solution like (2, −1).

    Answer

  4. Question 4

    To solve y = −3x + 1 and −2x + 3y = −9 by substitution, which is the correct next line?

    • A. −2x − 3(−3x + 1) = −9
    • B. −2x + 3(−3x + 1) = −9
    • C. −3x + 1 + 3y = −9
    • D. −2x + 3·−3x + 1 = −9

  5. Question 5

    Solve the system: y = x + 10 3x − 4y = −36 Type the solution like (2, −1).

    Answer

  6. Question 6

    Solve the system: x = −3y − 9 −x − 5y = 17 Type the solution like (2, −1).

    Answer

  7. Question 7

    To solve y = x − 7 and −x − 3y = −13 by substitution, which is the correct next line?

    • A. −x − 3·x − 7 = −13
    • B. −x − 3(x − 7) = −13
    • C. x − 7 − 3y = −13
    • D. −x + 3(x − 7) = −13

  8. Question 8

    Solve the system: y = x − 7 4x − 2y = 22 Type the solution like (2, −1).

    Answer

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Answer key: Solve a linear system by substitution

  1. 1. D. 2x + 5(x − 1) = −6

  2. 2. (−4, −4)

  3. 3. (1, −1)

  4. 4. B. −2x + 3(−3x + 1) = −9

  5. 5. (−4, 6)

  6. 6. (3, −4)

  7. 7. B. −x − 3(x − 7) = −13

  8. 8. (4, −3)