Question
Match each equation with what it represents. Prompts (1) \( y=m x+b \) (2) \( y-y_{1}=m\left(x-x_{1}\right) \) (3) \( x=a \) Select match (4) \( y=b \) Select match Select match Last saved \( 12: 58: 18 \) PM
Ask by Klein Malone. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
1. \( y = m x + b \) — Slope-Intercept Form of a Line
2. \( y - y_{1} = m(x - x_{1}) \) — Point-Slope Form of a Line
3. \( x = a \) — Vertical Line
4. \( y = b \) — Horizontal Line
Solution
Certainly! Let's match each equation with what it represents:
1. **\( y = m x + b \)**
- **Represents:** *Slope-Intercept Form of a Line*
- **Explanation:** This equation expresses a straight line where \( m \) is the slope and \( b \) is the y-intercept (the point where the line crosses the y-axis).
2. **\( y - y_{1} = m(x - x_{1}) \)**
- **Represents:** *Point-Slope Form of a Line*
- **Explanation:** This form is used when you know the slope \( m \) of a line and a specific point \( (x_{1}, y_{1}) \) that the line passes through.
3. **\( x = a \)**
- **Represents:** *Vertical Line*
- **Explanation:** This equation describes a vertical line where every point on the line has an x-coordinate of \( a \). It is parallel to the y-axis.
4. **\( y = b \)**
- **Represents:** *Horizontal Line*
- **Explanation:** This equation describes a horizontal line where every point on the line has a y-coordinate of \( b \). It is parallel to the x-axis.
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**Summary of Matches:**
1. **\( y = m x + b \)** — *Slope-Intercept Form of a Line*
2. **\( y - y_{1} = m(x - x_{1}) \)** — *Point-Slope Form of a Line*
3. **\( x = a \)** — *Vertical Line*
4. **\( y = b \)** — *Horizontal Line*
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Simplify this solution Beyond the Answer
The equation \( y = mx + b \) represents the slope-intercept form of a linear equation. In this representation, \( m \) is the slope of the line, and \( b \) is the y-intercept, where the line crosses the y-axis. This provides a clear and concise way to graph a linear equation and understand its behavior! The equation \( x = a \) represents a vertical line on the Cartesian plane, where the line crosses the x-axis at a specific value \( a \). This means that for all points on this line, the x-coordinate remains constant at \( a \), while the y-coordinate can take on any value! It’s like saying, "Hey, I’m just standing here along this vertical path!"
