Answer
The equation \( y = x - 18 \) represents a straight line with a slope of 1 and a y-intercept at (0, -18). It crosses the y-axis at -18 and increases by 1 unit for every 1 unit increase in x.
Solution
The equation you provided is:
\[ y = x - 18 \]
This is the equation of a straight line in the Cartesian plane. Here's a breakdown of its components and what it represents:
### **1. Slope-Intercept Form**
The equation is in the **slope-intercept form**, which is generally written as:
\[ y = mx + b \]
Where:
- **\( m \)** is the **slope** of the line.
- **\( b \)** is the **y-intercept**, the point where the line crosses the y-axis.
### **2. Identifying Slope and Y-Intercept**
For the equation \( y = x - 18 \):
- **Slope (\( m \))**: The coefficient of \( x \) is **1**. This means the line rises **1 unit** vertically for every **1 unit** it moves horizontally to the right.
- **Y-Intercept (\( b \))**: The constant term is **-18**. This indicates that the line crosses the y-axis at the point \( (0, -18) \).
### **3. Graphical Representation**
To graph the equation:
1. **Plot the Y-Intercept**: Start by plotting the point \( (0, -18) \) on the y-axis.
2. **Use the Slope**: From the y-intercept, use the slope to find another point. Since the slope is 1, move **1 unit to the right** (positive direction of x) and **1 unit up** (positive direction of y) to reach the next point at \( (1, -17) \).
3. **Draw the Line**: Connect these points with a straight line extending in both directions.
### **4. Characteristics of the Line**
- **Slope (\( m = 1 \))**: Indicates a positive correlation; as \( x \) increases, \( y \) increases at the same rate.
- **Y-Intercept (\( b = -18 \))**: The line crosses the y-axis at \( -18 \).
- **Angle**: The line forms a 45-degree angle with both the x-axis and y-axis because the slope is 1 (rise over run is 1/1).
### **5. Applications and Context**
Without additional context, it's hard to determine the specific application of this equation. However, linear equations like this are commonly used in various fields such as:
- **Economics**: Representing cost, revenue, or profit lines.
- **Physics**: Describing relationships like distance over time at constant speed.
- **Everyday Problem Solving**: Such as converting temperatures or calculating expenses.
### **6. Example Problem**
**Find the value of \( y \) when \( x = 10 \).**
**Solution:**
\[ y = 10 - 18 = -8 \]
So, when \( x = 10 \), \( y = -8 \).
### **7. Additional Considerations**
If this equation is part of a larger problem (e.g., solving a system of equations, finding intersections, etc.), please provide more details so I can offer a more tailored explanation or solution.
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