Linear Solver Intervention Worksheet
Name: _________________________________ Date: _______________
Class / Period: ___________
Targeted Intervention Guide
Linear Equation Blueprint
Mastering equations step-by-step through balance and inverse operations
Page 1 of 3 Level 1: Two-Step Equations
The Golden Balance Rule
Whatever mathematical operation you do to one side of the equals sign, you must do to the other side.
Inverse Operations Key
\(+\) cancels \(-\) \(\times\) cancels \(\div\)
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Level 1: Two-Step Equations (\(ax + b = c\))
Strategy: Undo addition/subtraction first, then undo multiplication/division.
Step-by-Step Model (I Do) Solve: \(3x + 7 = 22\)
Step 1: Undo \(+7\)
Subtract 7 from both sides:
\(3x + 7 - 7 = 22 - 7\)
\(3x = 15\)
Step 2: Undo \(\cdot 3\)
Divide both sides by 3:
\(\frac{3x}{3} = \frac{15}{3}\)
\(x = 5\)
Step 3: Check Answer
Substitute 5 into original:
\(3(5) + 7 = 22\)
\(15 + 7 = 22\) ✓ True
Guided Practice (We Do) — Fill in the blanks Solve: \(4x - 5 = 19\)
Step 1: Undo \(-5\)
Add ______ to both sides:
\(4x - 5 + \_\_ = 19 + \_\_\)
\(4x =\) ______
Step 2: Undo \(\cdot 4\)
Divide both sides by ______:
\(\frac{4x}{\_\_} = \frac{\_\_}{\_\_}\)
\(x =\) ______
Step 3: Verification
Check in original equation:
\(4(\_\_) - 5 \stackrel{?}{=} 19\)
\(\_\_ - 5 = 19\)
Independent Practice (You Do) — Show all balancing steps Solve & verify
Problem 1: \(5x + 8 = 38\)
Show Step 1 (Undo \(+\)), Step 2 (Undo \(\cdot\)):
Check: \(5(\quad) + 8 = \) \(x =\) ______
Problem 2: \(\frac{x}{3} - 4 = 7\)
Show Step 1 (Undo \(-\)), Step 2 (Undo \(\div\)):
Check: \(\frac{(\quad)}{3} - 4 = \) \(x =\) ______
Name: _________________________________ Date: _______________
Page 2 of 3 • Level 2: Variables on Both Sides
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Level 2: Variables on Both Sides (\(ax + b = cx + d\))
The Variable Gathering Rule: Move all variable terms to one side of '=' and all constants to the other side. Pro-Tip: Eliminate the smaller variable term first to keep your variable coefficient positive!
Step-by-Step Model (I Do) Solve: \(7x - 4 = 3x + 16\)
Step 1: Move Variable
Subtract \(3x\) on both sides:
\(7x \mathbf{-3x} - 4 = 3x \mathbf{-3x} + 16\)
\(4x - 4 = 16\)
Step 2: Move Constant
Add 4 to both sides:
\(4x - 4 \mathbf{+ 4} = 16 \mathbf{+ 4}\)
\(4x = 20\)
Step 3: Isolate \(x\)
Divide both sides by 4:
\(\frac{4x}{4} = \frac{20}{4}\)
\(x = 5\)
Step 4: Check Both
Test \(x = 5\) on both sides:
\(7(5) - 4 = 31\)
\(3(5) + 16 = 31\) ✓
Guided Practice (We Do) — Follow the scaffolding Solve: \(6x + 5 = 2x + 29\)
Step 1: Move Variable
Subtract \(2x\) from both sides:
\(6x - \_\_x + 5 = 29\)
\(\_\_x + 5 = 29\)
Step 2: Move Constant
Subtract 5 from both sides:
\(\_\_x = 29 - 5\)
\(\_\_x = \_\_\)
Step 3: Divide
Divide by coefficient:
\(x = \frac{\_\_}{\_\_}\)
\(x =\) ______
Step 4: Check
Substitute your answer:
\(6(\_\_) + 5 = \_\_\)
\(2(\_\_) + 29 = \_\_\)
Independent Practice (You Do) — Show each balanced step Collect terms & isolate
Problem 3: \(8x - 7 = 3x + 18\)
Show Step 1 (Move \(x\)), Step 2 (Move \(\#\)), Step 3 (Divide):
Check Left = Right: \(x =\) ______
Problem 4: \(5x + 12 = 9x - 8\)
Show Step 1 (Move \(x\)), Step 2 (Move \(\#\)), Step 3 (Divide):
Check Left = Right: \(x =\) ______
Name: _________________________________ Date: _______________
Page 3 of 3 • Level 3: Multi-Step & Analysis
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Level 3: Multi-Step & Distributive Property
Step-by-Step Model (I Do) Solve: \(3(2x - 4) + 5 = 29\)
Step 1: Distribute
Multiply 3 across:
\(3 \cdot 2x - 3 \cdot 4 + 5 = 29\)
\(6x - 12 + 5 = 29\)
Step 2: Combine Terms
Add \(-12 + 5\):
\(-12 + 5 = -7\)
\(6x - 7 = 29\)
Step 3: Solve 2-Step
Add 7, then divide by 6:
\(6x = 36\)
\(x = 6\)
Step 4: Check
Substitute \(x = 6\):
\(3(2(6) - 4) + 5\)
\(3(8) + 5 = 29\) ✓
Guided Practice (We Do) Solve: \(4(x + 3) - 7 = 21\)
Step 1: Distribute & Combine
Multiply 4 to both inside terms:
\(4x + \_\_ - 7 = 21\)
\(4x +\) ______ \(= 21\)
Step 2: Isolate Variable
Undo addition and multiplication:
\(4x = 21 - \_\_\)
\(4x =\) ______
Step 3: Solution & Check
Divide and substitute:
\(x =\) ______
\(4(\_\_ + 3) - 7 \stackrel{?}{=} 21\)
Intervention Lab: Error Detective Find & Fix the Classic Mistake
Student's Incorrect Work:
Solve: \(5(x - 2) = 35\)
Line 1: \(5x - 2 = 35\)
Line 2: \(5x = 37\)
Line 3: \(x = 7.4\)
1. Explain what mistake was made in Line 1:
2. Provide the correct step-by-step solution:
Mastery Challenge (You Do) Solve: \(2(3x - 1) = 4x + 10\)
Show full distribution, collecting variables, and isolating \(x\):
Confidence Check: ◯ Need Help ◯ Getting There ◯ I Got This!
Solution: \(x =\) ______
