Multi-Step Equations

When working with multi-step equations, the goal is to isolate the variable on one side of the equation to find its value. You may encounter equations that involve addition, subtraction, multiplication, and division, sometimes in combination. In this article, we will walk through the various steps and strategies to effectively solve multi-step equations, complete with examples and practice problems to enhance your understanding.

Understanding the Components

Before we dive into the solving process, let’s refresh our understanding of the key elements involved in equations:

  1. Variables: Letters representing unknown values, such as \( x \), \( y \), or \( z \).
  2. Constants: Fixed numerical values in an equation.
  3. Operators: Symbols that represent operations, such as \( + \), \( - \), \( \times \), or \( \div \).
  4. The Equals Sign \( = \): Indicates that the two sides of the equation are equal in value.

The Order of Operations

One crucial aspect of solving equations correctly is following the proper order of operations, often remembered by the acronym PEMDAS:

  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

While the order of operations is typically applied in expressions, it also aids in solving equations when you need to simplify both sides.

Steps to Solve Multi-Step Equations

Step 1: Simplify Each Side of the Equation

Begin by simplifying each side of the equation if necessary. This could involve combining like terms or applying the distributive property.

Example: \[ 3(x + 2) + 4 = 28 \]

Using the distributive property: \[ 3x + 6 + 4 = 28 \]

Now simplify: \[ 3x + 10 = 28 \]

Step 2: Eliminate the Constant on One Side

Next, isolate terms involving the variable by eliminating constants on one side. To do this, you can either add or subtract the constant from both sides of the equation.

Example: From the previous equation: \[ 3x + 10 = 28 \]

Subtract 10 from both sides: \[ 3x = 18 \]

Step 3: Isolate the Variable

After simplifying, isolate the variable by performing the opposite operation (multiplication or division) of the coefficient on the variable.

Example: Now, to isolate \( x \) in the equation \( 3x = 18 \), divide both sides by 3: \[ x = 6 \]

Step 4: Check Your Solution

Lastly, always plug your solution back into the original equation to ensure that both sides are equal. This step is crucial to verify that there were no mistakes in calculations.

Check: Substituting \( x = 6\) back into the original equation: \[ 3(6 + 2) + 4 = 28 \]
\[ 3(8) + 4 = 28 \]
\[ 24 + 4 = 28 \]
\[ 28 = 28 \]
The solution checks out!

More Examples

Let’s go through a couple more examples to solidify your understanding.

Example 1

Solve the equation: \[ 2(x - 3) + 4 = 10 \]

Step 1: Distribute the 2: \[ 2x - 6 + 4 = 10 \]

Step 2: Gather like terms: \[ 2x - 2 = 10 \]

Step 3: Add 2 to both sides: \[ 2x = 12 \]

Step 4: Divide by 2: \[ x = 6 \]

Check: \[ 2(6 - 3) + 4 = 10 \]
\[ 2(3) + 4 = 10 \]
\[ 6 + 4 = 10 \]
\[ 10 = 10 \]

Example 2

Solve the equation: \[ 5(x + 2) - 3 = 2x + 9 \]

Step 1: Distribute the 5: \[ 5x + 10 - 3 = 2x + 9 \]

Simplify: \[ 5x + 7 = 2x + 9 \]

Step 2: Subtract \( 2x \) from both sides: \[ 3x + 7 = 9 \]

Step 3: Subtract 7 from both sides: \[ 3x = 2 \]

Step 4: Divide by 3: \[ x = \frac{2}{3} \]

Check: Substituting back: \[ 5\left(\frac{2}{3} + 2\right) - 3 = 2\left(\frac{2}{3}\right) + 9 \]
Calculating: \[ 5\left(\frac{2}{3} + \frac{6}{3}\right) - 3 = \frac{4}{3} + 9 \]
\[ 5\left(\frac{8}{3}\right) - 3 = \frac{4}{3} + \frac{27}{3} \]
\[ \frac{40}{3} - \frac{9}{3} = \frac{31}{3} \]
\[ \frac{31}{3} = \frac{31}{3} \]

Practice Problems

Now it’s your turn! Try solving these multi-step equations:

  1. \( 4(x + 1) - 6 = 2x + 2 \)
  2. \( 3x + 5(2 - x) = 8 \)
  3. \( 7 - 2(3x - 4) = x + 5 \)

Problems with Solutions

Once you’ve tried working through those, here are the solutions for you to check your work:

  1. \( x = 0 \)
  2. \( x = 1 \)
  3. \( x = -2 \)

Conclusion

Mastering multi-step equations opens the door to more advanced algebra techniques, which are essential for success in mathematics. By following the steps outlined in this article and practicing regularly, you’ll build confidence in your ability to solve complex equations. Remember, the key is to simplify step by step, isolate the variable, and always check your work. Happy solving!