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Math 8 & 9 Linear Equations

 

1. What is a Two-Step Equation?

A two-step equation requires two operations to isolate the variable (get $x$ by itself).

  • Example: 3x + 5 = 20

  • The Goal: Reverse the operations applied to $x$ using the Order of Operations in REVERSE (SAMDEB).

2. The Golden Rules of Solving

  1. Balance: Whatever you do to one side of the equal sign, you MUST do to the other.

  2. Undo Addition/Subtraction first: Get rid of the constant (the number without a variable).

  3. Undo Multiplication/Division second: Get rid of the coefficient (the number attached to the variable).

Example Walkthrough: 2x - 7 = 11

  • Step 1 (Add/Subtract): Add 7 to both sides to cancel out the -7.

    • 2x - 7 + 7 = 11 + 7 

    • 2x = 18

  • Step 2 (Multiply/Divide): Divide both sides by 2 to cancel out the multiplication.

    • 2x / 2 = 18 / 2 

    • x = 9

3. Modeling Word Problems (The "Let" Statement)

To solve a word problem, you must translate English into Math.

Common Keywords

  • Addition: Sum, increased by, more than, plus, total.

  • Subtraction: Difference, decreased by, less than, minus.

  • Multiplication: Product, times, per, each, of.

  • Division: Quotient, split, per.

  • Equals: Is, results in, totals.

The "Template" Equation

Most word problems follow this format:

(Rate × Variable) + Starting Amount = Total

Example: A taxi charges a flat fee of $5 plus $2 per kilometer. If the total bill was $21, how far did you travel?

  • Let k = number of kilometers.

  • Equation: 2k + 5 = 21

  • Solve: 2k = 16 

  • k = 8 km

4. PRACTICE PROBLEMS: Linear Equations

Section 1: Solving Equations

Solve for the variable. Show your steps!

  1. 4x + 8 = 32

  2. 5n - 12 = 13

  3. x / 3 + 4 = 10

  4. 15 - 2y = 7

  5. -3w + 6 = -12

Section 2: Modeling Word Problems

Write the equation first, then solve.

  1. The Gym Membership: A gym charges a $50 sign-up fee and $20 per month. If Sarah has spent $250 in total, how many months has she been a member?

  2. The Cell Phone Plan: A plan costs $35 a month plus $0.10 per text message. If the bill is $42.50, how many texts were sent?

  3. The Number Mystery: Triple a number, then subtract 7. The result is 14. What is the number?

5. ANSWER KEY (Self-Check)

  1. x = 6 (4x = 24   ->   24/4)

  2. n = 5 (5n = 25   ->   25/5)

  3. x = 18 (x/3 = 6   ->   6 \times 3)

  4. y = 4 (-2y = -8   ->   -8/-2)

  5. w = 6 (-3w = -18   ->   -18/-3)

  6. 10 months (20m + 50 = 250   ->   20m = 200)

  7. 75 texts (0.10t + 35 = 42.50   ->   0.10t = 7.50)

  8. 7 (3n - 7 = 14   ->   3n = 21)

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