Question

What is an example of step equation?

Answer 1

A step equation is essentially a multi-step algebraic equation that requires several steps to solve. To solve it, you usually need to simplify expressions, combine like terms, and isolate the variable. Here's a detailed example to illustrate:

### Example Equation

Let's solve the following step equation:

\[ 3(x - 4) + 2 = 5x - 6 \]

### Steps to Solve the Equation

1. **Distribute the 3**:
   
   The equation begins with a term \( 3(x - 4) \). Apply the distributive property, which means multiplying 3 by each term inside the parentheses:

   \[ 3(x - 4) = 3 \cdot x - 3 \cdot 4 = 3x - 12 \]

   So the equation now is:

   \[ 3x - 12 + 2 = 5x - 6 \]

2. **Combine Like Terms**:

   On the left side of the equation, combine the constants \(-12\) and \(2\):

   \[ -12 + 2 = -10 \]

   This simplifies the equation to:

   \[ 3x - 10 = 5x - 6 \]

3. **Isolate the Variable**:

   To solve for \(x\), first get all the \(x\) terms on one side of the equation and the constants on the other side.

   - Subtract \(3x\) from both sides to move the \(x\) terms to one side:

     \[ 3x - 10 - 3x = 5x - 6 - 3x \]

     Simplifying this gives:

     \[ -10 = 2x - 6 \]

   - Next, add \(6\) to both sides to move the constants to one side:

     \[ -10 + 6 = 2x - 6 + 6 \]

     Simplifying this gives:

     \[ -4 = 2x \]

4. **Solve for \(x\)**:

   Divide both sides by \(2\) to isolate \(x\):

   \[ \frac{-4}{2} = \frac{2x}{2} \]

   This simplifies to:

   \[ -2 = x \]

   So the solution to the equation is:

   \[ x = -2 \]

### Summary

To solve the step equation \( 3(x - 4) + 2 = 5x - 6 \), you need to:

1. Distribute the \(3\) through the parentheses.
2. Combine like terms on the left side.
3. Isolate \(x\) by moving all \(x\) terms to one side and constants to the other.
4. Solve for \(x\) by dividing.

This process demonstrates how to work through a multi-step algebraic equation, ensuring each step is clear and logically follows from the previous one.

Answer 2

A step equation, also known as a piecewise function or a step function, is a mathematical function that consists of different expressions or formulas for different intervals of the input variable. The function "steps" from one value to another, making it look like a staircase graph.

Here’s a basic example of a step function:

\[ f(x) = \begin{cases}
2 & \text{for } x < 0 \\
5 & \text{for } 0 \leq x < 3 \\
7 & \text{for } x \geq 3
\end{cases} \]

In this example:

- For \( x < 0 \), \( f(x) = 2 \)
- For \( 0 \leq x < 3 \), \( f(x) = 5 \)
- For \( x \geq 3 \), \( f(x) = 7 \)

The graph of this function would have horizontal line segments at \( y = 2 \), \( y = 5 \), and \( y = 7 \), with jumps at \( x = 0 \) and \( x = 3 \).