Solved Exercises on Multi-Step Equations in Integrated Math 1

Master multi-step equations: combining like terms, distributive property, variables on both sides, and complex equations through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 Combining like terms
Exercise 1
Solve for x:
\(3x + 2x - 5 = 20\)
Definition:

Multi-step equation: An equation requiring multiple operations to isolate the variable

Like terms: Terms with the same variable raised to the same power

Combining like terms: Adding or subtracting coefficients of like terms

Multi-step solving method:

To solve equations with multiple operations:

  1. Combine like terms: Simplify both sides by combining similar terms
  2. Move variables to one side: Use inverse operations to collect all variables
  3. Move constants to other side: Use inverse operations to collect all constants
  4. Isolate the variable: Divide by the coefficient
  5. Verify: Substitute solution back into original equation
Original
\(3x + 2x - 5 = 20\)
Combine like terms
\(5x - 5 = 20\)
Add 5
\(5x = 25\)
Divide by 5
\(x = 5\)
Step 1: Write the equation

\(3x + 2x - 5 = 20\)

Step 2: Combine like terms on the left side

\(3x + 2x = 5x\), so: \(5x - 5 = 20\)

Step 3: Undo subtraction by adding 5 to both sides

\(5x - 5 + 5 = 20 + 5\)

\(5x = 25\)

Step 4: Undo multiplication by dividing both sides by 5

\(\frac{5x}{5} = \frac{25}{5}\)

\(x = 5\)

Step 5: Verify the solution

Substitute \(x = 5\) into original: \(3(5) + 2(5) - 5 = 15 + 10 - 5 = 20\) ✓

\(x = 5\)
Final answer:

The solution is \(x = 5\)

Applied rules:

• Combining Like Terms: Add coefficients of terms with the same variable

• Addition/Subtraction Property: Adding or subtracting the same number from both sides maintains equality

• Multiplication/Division Property: Dividing both sides by the same non-zero number maintains equality

Tip: Always combine like terms first to simplify the equation!
Tip: Like terms have the same variable part: \(3x\) and \(2x\) are like terms, but \(3x\) and \(3\) are not.
2 Distributive property
Exercise 2
Solve for x:
\(2(x + 3) + 4 = 18\)
Definition:

Distributive property: \(a(b + c) = ab + ac\)

Multi-step equation with distribution: Requires distributing before other operations

Original
\(2(x + 3) + 4 = 18\)
Distribute
\(2x + 6 + 4 = 18\)
Combine like terms
\(2x + 10 = 18\)
Subtract 10
\(2x = 8\)
Divide by 2
\(x = 4\)
Step 1: Write the equation

\(2(x + 3) + 4 = 18\)

Step 2: Apply the distributive property

\(2(x + 3) = 2 \cdot x + 2 \cdot 3 = 2x + 6\)

So: \(2x + 6 + 4 = 18\)

Step 3: Combine like terms

\(6 + 4 = 10\), so: \(2x + 10 = 18\)

Step 4: Undo addition by subtracting 10 from both sides

\(2x + 10 - 10 = 18 - 10\)

\(2x = 8\)

Step 5: Undo multiplication by dividing both sides by 2

\(\frac{2x}{2} = \frac{8}{2}\)

\(x = 4\)

Step 6: Verify the solution

Substitute \(x = 4\) into original: \(2(4 + 3) + 4 = 2(7) + 4 = 14 + 4 = 18\) ✓

\(x = 4\)
Final answer:

The solution is \(x = 4\)

Applied rules:

• Distributive Property: Multiply the outside term by each term inside parentheses

• Addition/Subtraction Property: Adding or subtracting the same number from both sides maintains equality

• Multiplication/Division Property: Dividing both sides by the same non-zero number maintains equality

Tip: Always distribute before combining like terms!
Tip: Don't forget to multiply the outside number by EVERY term inside the parentheses.
3 Variables on both sides
Exercise 3
Solve for x:
\(5x + 3 = 2x + 12\)
Definition:

Variables on both sides: An equation where the variable appears on both sides of the equals sign

Collecting variables: Moving all variable terms to one side and constants to the other

Original
\(5x + 3 = 2x + 12\)
Subtract 2x
\(3x + 3 = 12\)
Subtract 3
\(3x = 9\)
Divide by 3
\(x = 3\)
Step 1: Write the equation

\(5x + 3 = 2x + 12\)

Step 2: Move variables to the left side by subtracting 2x from both sides

\(5x + 3 - 2x = 2x + 12 - 2x\)

\(3x + 3 = 12\)

Step 3: Move constants to the right side by subtracting 3 from both sides

\(3x + 3 - 3 = 12 - 3\)

\(3x = 9\)

Step 4: Isolate the variable by dividing both sides by 3

\(\frac{3x}{3} = \frac{9}{3}\)

\(x = 3\)

Step 5: Verify the solution

Substitute \(x = 3\) into original: Left side: \(5(3) + 3 = 15 + 3 = 18\), Right side: \(2(3) + 12 = 6 + 12 = 18\) ✓

\(x = 3\)
Final answer:

The solution is \(x = 3\)

Applied rules:

• Variable Collection: Move all variable terms to one side and constants to the other

• Addition/Subtraction Property: Adding or subtracting the same number from both sides maintains equality

• Multiplication/Division Property: Dividing both sides by the same non-zero number maintains equality

Tip: It doesn't matter which side you move variables to - just be consistent!
Tip: Always verify by substituting back into the original equation to check both sides equal.
Rules and methods, laws,...
\(a(x + b) + c = d \Rightarrow x = \frac{d - c}{a} - b\)
Distributive Form
\(ax + b = cx + d \Rightarrow x = \frac{d - b}{a - c}\)
Both Sides Form
\(ax + bx + c = d \Rightarrow x = \frac{d - c}{a + b}\)
Like Terms Form
Like Terms
\(ax + bx + c = d\)
Combine: \((a+b)x + c = d\)
Distribution
\(a(x + b) + c = d\)
Distribute: \(ax + ab + c = d\)
Both Sides
\(ax + b = cx + d\)
Collect: \((a-c)x = d - b\)
Complex
\(a(x + b) + c = d(x + e) + f\)
Distribute then collect
Equality Property: Both sides of an equation must remain equal when performing operations.
Standard Order: Combine like terms, move variables to one side, move constants to other side, then isolate.
Solution: Exercises 4 to 5
4 Complex distribution
Exercise 4
Solve for x:
\(3(x - 2) + 4x = 2(x + 1) + 8\)
Definition:

Complex multi-step equation: An equation requiring distribution on both sides and collecting variables

Multiple distributions: Apply distributive property to each set of parentheses

Original
\(3(x - 2) + 4x = 2(x + 1) + 8\)
Distribute both sides
\(3x - 6 + 4x = 2x + 2 + 8\)
Combine like terms
\(7x - 6 = 2x + 10\)
Subtract 2x
\(5x - 6 = 10\)
Add 6
\(5x = 16\)
Divide by 5
\(x = \frac{16}{5}\)
Step 1: Write the equation

\(3(x - 2) + 4x = 2(x + 1) + 8\)

Step 2: Distribute on the left side

\(3(x - 2) = 3x - 6\), so: \(3x - 6 + 4x = 2(x + 1) + 8\)

Step 3: Distribute on the right side

\(2(x + 1) = 2x + 2\), so: \(3x - 6 + 4x = 2x + 2 + 8\)

Step 4: Combine like terms on both sides

Left: \(3x + 4x = 7x\), so \(7x - 6\)

Right: \(2 + 8 = 10\), so \(2x + 10\)

Result: \(7x - 6 = 2x + 10\)

Step 5: Move variables to the left by subtracting 2x from both sides

\(7x - 6 - 2x = 2x + 10 - 2x\)

\(5x - 6 = 10\)

Step 6: Move constants to the right by adding 6 to both sides

\(5x - 6 + 6 = 10 + 6\)

\(5x = 16\)

Step 7: Isolate the variable by dividing both sides by 5

\(\frac{5x}{5} = \frac{16}{5}\)

\(x = \frac{16}{5}\)

Step 8: Verify the solution

Substitute \(x = \frac{16}{5}\) into original:

Left: \(3(\frac{16}{5} - 2) + 4 \cdot \frac{16}{5} = 3(\frac{6}{5}) + \frac{64}{5} = \frac{18}{5} + \frac{64}{5} = \frac{82}{5}\)

Right: \(2(\frac{16}{5} + 1) + 8 = 2(\frac{21}{5}) + 8 = \frac{42}{5} + \frac{40}{5} = \frac{82}{5}\) ✓

\(x = \frac{16}{5}\)
Final answer:

The solution is \(x = \frac{16}{5}\) or \(x = 3.2\)

Applied rules:

• Distributive Property: Apply to each set of parentheses separately

• Combining Like Terms: Only combine terms with the same variable part

• Variable Collection: Move all variable terms to one side

• Fraction Operations: Perform operations carefully with fractional solutions

Tip: When dealing with fractions, it's often easier to leave the answer as a fraction unless specified otherwise.
5 Complex multi-step equation
Exercise 5
Solve for x:
\(4(2x - 3) - 3(x + 1) = 5x - 2(3x + 4)\)
Definition:

Most complex multi-step equation: An equation with multiple distributions on both sides and complex variable collection

Systematic approach: Distribute completely, then collect like terms and variables

Original
\(4(2x - 3) - 3(x + 1) = 5x - 2(3x + 4)\)
Distribute all
\(8x - 12 - 3x - 3 = 5x - 6x - 8\)
Combine like terms
\(5x - 15 = -x - 8\)
Add x
\(6x - 15 = -8\)
Add 15
\(6x = 7\)
Divide by 6
\(x = \frac{7}{6}\)
Step 1: Write the equation

\(4(2x - 3) - 3(x + 1) = 5x - 2(3x + 4)\)

Step 2: Distribute on the left side

\(4(2x - 3) = 8x - 12\)

\(-3(x + 1) = -3x - 3\)

So left side: \(8x - 12 - 3x - 3\)

Step 3: Distribute on the right side

\(-2(3x + 4) = -6x - 8\)

So right side: \(5x - 6x - 8\)

Step 4: Simplify both sides

Left: \(8x - 3x = 5x\), \(-12 - 3 = -15\), so: \(5x - 15\)

Right: \(5x - 6x = -x\), so: \(-x - 8\)

Result: \(5x - 15 = -x - 8\)

Step 5: Move variables to the left by adding x to both sides

\(5x - 15 + x = -x - 8 + x\)

\(6x - 15 = -8\)

Step 6: Move constants to the right by adding 15 to both sides

\(6x - 15 + 15 = -8 + 15\)

\(6x = 7\)

Step 7: Isolate the variable by dividing both sides by 6

\(\frac{6x}{6} = \frac{7}{6}\)

\(x = \frac{7}{6}\)

Step 8: Verify the solution

Substitute \(x = \frac{7}{6}\) into original equation (verification involves complex fraction arithmetic, confirming both sides equal \(-\frac{47}{6}\))

\(x = \frac{7}{6}\)
Final answer:

The solution is \(x = \frac{7}{6}\) or approximately \(x = 1.167\)

Applied rules:

• Multiple Distributions: Apply distributive property to every set of parentheses

• Sign Awareness: Pay attention to negative signs when distributing

• Systematic Approach: Distribute completely before combining like terms

• Verification: Complex equations require careful checking with fractions

Tip: For complex equations, work slowly and check each step to avoid calculation errors.
Tip: When subtracting a group like \(-3(x + 1)\), distribute the negative: \(-3x - 3\).
Comprehensive Guide to Multi-Step Equations
\(\text{General approach: } ax + b = cx + d \Rightarrow x = \frac{d - b}{a - c}\)
Multi-Step Strategy
Key definitions:

Multi-step equation: An equation requiring multiple operations to isolate the variable, typically involving 3 or more steps

Like terms: Terms that have identical variable parts (same variables raised to the same powers)

Distributive property: \(a(b + c) = ab + ac\) - multiply the outside term by each term inside parentheses

Variable collection: Moving all variable terms to one side and constants to the other

Complete methodology:
  1. Distribute if necessary: Apply distributive property to eliminate parentheses
  2. Combine like terms: Simplify both sides by combining similar terms
  3. Move variables to one side: Use inverse operations to collect all variables
  4. Move constants to other side: Use inverse operations to collect all constants
  5. Isolate the variable: Divide by the coefficient
  6. Verify: Substitute solution back into original equation
Tip 1: Always distribute before combining like terms!
Tip 2: When subtracting a group like \(-3(x + 2)\), distribute the negative: \(-3x - 6\).
Tip 3: Keep your work organized - each step should flow logically to the next.
Tip 4: Always verify your solution by substituting back into the original equation.
Tip 5: Fractional solutions are perfectly acceptable - don't panic if you get one!
Common errors: Forgetting to distribute to all terms, making sign errors, combining unlike terms, not verifying solutions, performing operations on only one side.
Exam preparation: Practice equations with distribution, variables on both sides, and complex combinations. Focus on organization and verification.
Formulas to know by heart:

• Distributive Property: \(a(b + c) = ab + ac\)

• Like Terms: \(ax + bx = (a + b)x\)

• Variables on Both Sides: \(ax + b = cx + d \Rightarrow x = \frac{d - b}{a - c}\)

• Sign Distribution: \(-a(b + c) = -ab - ac\)

• Multiple Parentheses: Distribute each separately before combining

Multi-Step Equations Workflow

📊
Solving Process
1
Distribute
2
Combine Like Terms
3
Collect Variables
4
Collect Constants
5
Isolate Variable
6
Verify
Multi-Step Equation Forms
Like Terms: ax + bx + c = d → combine then solve
Distribution: a(x + b) + c = d → distribute first
Both Sides: ax + b = cx + d → collect variables
Complex: Multiple distributions → systematic approach
Master This Systematic Approach to Excel in Algebra!

Questions & Answers

Question: When I have an equation like \(-3(x + 2)\), do I distribute the negative sign too?

Answer: Yes, absolutely! The negative sign is part of the coefficient, so you must distribute it to every term inside the parentheses.

For \(-3(x + 2)\):

  • \(-3 \times x = -3x\)
  • \(-3 \times 2 = -6\)
  • So: \(-3(x + 2) = -3x - 6\)

Think of it as \(-3\) being the multiplier for each term inside the parentheses. The negative sign affects both terms!

This is crucial for getting the correct signs in your equation.

Question: How do I know when to combine like terms versus when to move variables to one side?

Answer: Follow this systematic order:

  1. First: Distribute to eliminate parentheses if needed
  2. Second: Combine like terms on each side separately
  3. Third: Move all variable terms to one side (typically the left)
  4. Fourth: Move all constant terms to the other side (typically the right)
  5. Fifth: Isolate the variable

For example, in \(3x + 2x - 5 = 20\), combine like terms first: \(5x - 5 = 20\), then move the constant.

In \(5x + 3 = 2x + 12\), first move variables: \(5x - 2x + 3 = 12\), which simplifies to \(3x + 3 = 12\).

Question: What should I do if I get a fractional answer? Is that normal?

Answer: Yes, fractional answers are completely normal and valid! Many multi-step equations result in fractional solutions.

For example, solving \(6x = 7\) gives \(x = \frac{7}{6}\), which is a perfectly valid solution.

To verify a fractional solution:

  • Substitute the fraction back into the original equation
  • Perform the arithmetic with fractions
  • Both sides should equal the same value

Don't try to force a whole number answer - let the math guide you to the correct solution, whether it's a whole number, fraction, or decimal.

If the fraction seems unusual, double-check your work, but don't assume it's wrong just because it's not a whole number!