Step By Step Solving Multi Step Equations

Solving multi-step equations is a fundamental skill in algebra, building upon the basic principles of solving one-step and two-step equations. Mastering this skill is crucial for success in more advanced mathematical topics. Multi-step equations involve more than two operations and often require simplifying expressions by combining like terms or using the distributive property before isolating the variable. This comprehensive guide provides a step-by-step approach to solving multi-step equations, complete with examples and explanations to enhance your understanding.

Understanding Multi-Step Equations

Multi-step equations are algebraic equations that require several steps to solve. These equations typically involve:

• Variables: Symbols (usually letters) representing unknown values.
• Constants: Numbers that have a fixed value.
• Coefficients: Numbers multiplied by variables.
• Operations: Addition, subtraction, multiplication, and division.
• Parentheses: Grouping symbols indicating an order of operations.

The goal in solving multi-step equations is to isolate the variable on one side of the equation to determine its value. This involves performing inverse operations to "undo" the operations applied to the variable.

Step-by-Step Guide to Solving Multi-Step Equations

Step 1: Simplify Both Sides of the Equation

Before you start isolating the variable, simplify each side of the equation as much as possible. This may involve:

• Distributive Property: If there are parentheses, distribute any coefficients to the terms inside the parentheses.
• Combining Like Terms: Combine any like terms on the same side of the equation.

Distributive Property

The distributive property states that a( b + c ) = ab + ac. Use this property to remove parentheses by multiplying the term outside the parentheses by each term inside.

Example:

Solve: 2(x + 3) = 10

1. Distribute: Multiply 2 by both x and 3.

   2x + 2(3) = 10

   2x + 6 = 10

2. Simplify: The equation is now simplified and ready for the next steps.

Combining Like Terms

Like terms are terms that have the same variable raised to the same power. Combine like terms by adding or subtracting their coefficients.

Example:

Solve: 3x + 2 + 5x - 1 = 15

1. Identify Like Terms: In this equation, 3x and 5x are like terms, and 2 and -1 are like terms.

2. Combine Like Terms: Add 3x and 5x to get 8x. Add 2 and -1 to get 1.

   8x + 1 = 15

3. Simplify: The equation is now simplified and ready for the next steps.

Step 2: Isolate the Variable Term

After simplifying both sides of the equation, the next step is to isolate the variable term. This means getting the term with the variable alone on one side of the equation.

To do this, use inverse operations to eliminate any constants that are being added or subtracted from the variable term. Remember, whatever operation you perform on one side of the equation, you must perform on the other side to maintain equality.

Example:

Solve: 8x + 1 = 15

1. Subtract 1 from Both Sides: To isolate the variable term 8x, subtract 1 from both sides of the equation.

   8x + 1 - 1 = 15 - 1

   8x = 14

2. Simplify: The variable term is now isolated on one side of the equation.

Step 3: Solve for the Variable

Once the variable term is isolated, the final step is to solve for the variable itself. This involves using inverse operations to eliminate any coefficients that are multiplying or dividing the variable.

To do this, divide both sides of the equation by the coefficient of the variable. This will leave the variable alone on one side of the equation, with its value on the other side.

Example:

Solve: 8x = 14

1. Divide Both Sides by 8: To solve for x, divide both sides of the equation by 8.

   8x/8 = 14/8

   x = 14/8

2. Simplify: Simplify the fraction to find the value of x.

   x = 7/4 or 1.75

Step 4: Check Your Solution

After finding a solution, it's always a good idea to check your answer by substituting it back into the original equation. If the equation holds true, then your solution is correct.

Example:

Check: 2(x + 3) = 10, where x = 2

1. Substitute: Replace x with 2 in the original equation.

   2(2 + 3) = 10

2. Simplify: Evaluate the expression.

   2(5) = 10

   10 = 10

3. Verify: Since the equation holds true, x = 2 is the correct solution.

Advanced Techniques for Solving Multi-Step Equations

Equations with Variables on Both Sides

When an equation has variables on both sides, the first step is to move all variable terms to one side and all constant terms to the other side. This is done by adding or subtracting terms from both sides of the equation.

Example:

Solve: 5x + 3 = 2x + 9

1. Subtract 2x from Both Sides: To move the variable terms to the left side, subtract 2x from both sides.

   5x + 3 - 2x = 2x + 9 - 2x

   3x + 3 = 9

2. Subtract 3 from Both Sides: To move the constant terms to the right side, subtract 3 from both sides.

   3x + 3 - 3 = 9 - 3

   3x = 6

3. Divide Both Sides by 3: To solve for x, divide both sides by 3.

   3x/3 = 6/3

   x = 2

Equations with Fractions

Equations with fractions can be more challenging to solve. One common strategy is to eliminate the fractions by multiplying both sides of the equation by the least common denominator (LCD) of all the fractions.

Example:

Solve: x/2 + 1/3 = 5/6

1. Find the LCD: The least common denominator of 2, 3, and 6 is 6.

2. Multiply Both Sides by the LCD: Multiply both sides of the equation by 6.

   6(x/2 + 1/3) = 6(5/6)

   6(x/2) + 6(1/3) = 6(5/6)

   3x + 2 = 5

3. Solve the Equation: Now solve the equation as usual.

   3x + 2 - 2 = 5 - 2

   3x = 3

   x = 1

Equations with Decimals

Equations with decimals can also be simplified by multiplying both sides of the equation by a power of 10 to eliminate the decimals.

Example:

Solve: 0.2x + 0.5 = 1.3

1. Multiply Both Sides by 10: Multiply both sides of the equation by 10 to eliminate the decimals.

   10(0.2x + 0.5) = 10(1.3)

   2x + 5 = 13

2. Solve the Equation: Now solve the equation as usual.

   2x + 5 - 5 = 13 - 5

   2x = 8

   x = 4

Common Mistakes to Avoid

• Incorrect Distribution: Ensure you distribute correctly by multiplying the term outside the parentheses by every term inside.
• Combining Unlike Terms: Only combine terms that have the same variable raised to the same power.
• Incorrect Inverse Operations: Use the correct inverse operations (addition/subtraction, multiplication/division) to isolate the variable.
• Forgetting to Apply Operations to Both Sides: Whatever operation you perform on one side of the equation, make sure to do it on the other side as well.
• Not Simplifying Completely: Always simplify both sides of the equation as much as possible before isolating the variable.

Examples of Solving Multi-Step Equations

Example 1:

Solve: 4(2x - 1) + 3 = 27

1. Distribute:

   4(2x) - 4(1) + 3 = 27

   8x - 4 + 3 = 27

2. Combine Like Terms:

   8x - 1 = 27

3. Add 1 to Both Sides:

   8x - 1 + 1 = 27 + 1

   8x = 28

4. Divide Both Sides by 8:

   8x/8 = 28/8

   x = 7/2 or 3.5

Example 2:

Solve: 3x + 5 = x - 7

1. Subtract x from Both Sides:

   3x + 5 - x = x - 7 - x

   2x + 5 = -7

2. Subtract 5 from Both Sides:

   2x + 5 - 5 = -7 - 5

   2x = -12

3. Divide Both Sides by 2:

   2x/2 = -12/2

   x = -6

Example 3:

Solve: x/3 - 1/2 = 1/6

1. Find the LCD: The least common denominator of 3, 2, and 6 is 6.

2. Multiply Both Sides by the LCD:

   6(x/3 - 1/2) = 6(1/6)

   6(x/3) - 6(1/2) = 6(1/6)

   2x - 3 = 1

3. Add 3 to Both Sides:

   2x - 3 + 3 = 1 + 3

   2x = 4

4. Divide Both Sides by 2:

   2x/2 = 4/2

   x = 2

Conclusion

Solving multi-step equations requires a systematic approach and careful attention to detail. By following the steps outlined in this guide, you can confidently tackle even the most complex equations. Remember to simplify both sides of the equation first, isolate the variable term, solve for the variable, and always check your solution. With practice and patience, you'll master this essential skill in algebra.