Answer :
Let's break down your question step-by-step and compare the given equations:
### 3. Compare the equation [tex]\( q = 2p + \frac{1}{3} \)[/tex] with [tex]\( y = mx + b \)[/tex]
In the general linear equation [tex]\( y = mx + b \)[/tex]:
- [tex]\( y \)[/tex] represents the dependent variable (output).
- [tex]\( x \)[/tex] represents the independent variable (input).
- [tex]\( m \)[/tex] is the slope of the line, indicating how steep the line is.
- [tex]\( b \)[/tex] is the y-intercept, representing the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
Now, let's compare this with the equation [tex]\( q = 2p + \frac{1}{3} \)[/tex]:
- [tex]\( q \)[/tex] represents the dependent variable, which is analogous to [tex]\( y \)[/tex].
- [tex]\( p \)[/tex] represents the independent variable, which is analogous to [tex]\( x \)[/tex].
- [tex]\( 2 \)[/tex] is the slope [tex]\( m \)[/tex].
- [tex]\( \frac{1}{3} \)[/tex] is the y-intercept [tex]\( b \)[/tex].
So, the equation [tex]\( q = 2p + \frac{1}{3} \)[/tex] can be interpreted in terms of the linear equation format as follows:
- Dependent variable ([tex]\( y \)[/tex] equivalent): [tex]\( q \)[/tex]
- Independent variable ([tex]\( x \)[/tex] equivalent): [tex]\( p \)[/tex]
- Slope ([tex]\( m \)[/tex]): [tex]\( 2 \)[/tex]
- Y-intercept ([tex]\( b \)[/tex]): [tex]\( \frac{1}{3} \)[/tex]
### 4. [tex]\( v = u \)[/tex]
The equation [tex]\( v = u \)[/tex] is a straightforward linear relationship where:
- [tex]\( v \)[/tex] and [tex]\( u \)[/tex] are variables that are equal to each other.
- There is no explicit slope or y-intercept in this equation since it's a direct equality.
- We can think of [tex]\( v = u \)[/tex] as a special case of the linear equation [tex]\( y = mx + b \)[/tex] where the slope [tex]\( m \)[/tex] is [tex]\( 1 \)[/tex] and the y-intercept [tex]\( b \)[/tex] is [tex]\( 0 \)[/tex].
Thus, the corresponding relationship is:
- Dependent variable ([tex]\( y \)[/tex] equivalent): [tex]\( v \)[/tex]
- Independent variable ([tex]\( x \)[/tex] equivalent): [tex]\( u \)[/tex]
- Slope ([tex]\( m \)[/tex]): [tex]\( 1 \)[/tex]
- Y-intercept ([tex]\( b \)[/tex]): [tex]\( 0 \)[/tex]
This indicates that for every unit increase in [tex]\( u \)[/tex], [tex]\( v \)[/tex] increases by the same amount. The line representing this relationship would be a 45-degree line passing through the origin on a graph with equal scales for [tex]\( u \)[/tex] and [tex]\( v \)[/tex].
In summary:
For the equation [tex]\( q = 2p + \frac{1}{3} \)[/tex]:
- Dependent variable: [tex]\( q \)[/tex]
- Independent variable: [tex]\( p \)[/tex]
- Slope [tex]\( m \)[/tex]: [tex]\( 2 \)[/tex]
- Y-intercept [tex]\( b \)[/tex]: [tex]\( \frac{1}{3} \)[/tex]
For the equation [tex]\( v = u \)[/tex]:
- Dependent variable: [tex]\( v \)[/tex]
- Independent variable: [tex]\( u \)[/tex]
- Slope [tex]\( m \)[/tex]: [tex]\( 1 \)[/tex]
- Y-intercept [tex]\( b \)[/tex]: [tex]\( 0 \)[/tex]
### 3. Compare the equation [tex]\( q = 2p + \frac{1}{3} \)[/tex] with [tex]\( y = mx + b \)[/tex]
In the general linear equation [tex]\( y = mx + b \)[/tex]:
- [tex]\( y \)[/tex] represents the dependent variable (output).
- [tex]\( x \)[/tex] represents the independent variable (input).
- [tex]\( m \)[/tex] is the slope of the line, indicating how steep the line is.
- [tex]\( b \)[/tex] is the y-intercept, representing the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
Now, let's compare this with the equation [tex]\( q = 2p + \frac{1}{3} \)[/tex]:
- [tex]\( q \)[/tex] represents the dependent variable, which is analogous to [tex]\( y \)[/tex].
- [tex]\( p \)[/tex] represents the independent variable, which is analogous to [tex]\( x \)[/tex].
- [tex]\( 2 \)[/tex] is the slope [tex]\( m \)[/tex].
- [tex]\( \frac{1}{3} \)[/tex] is the y-intercept [tex]\( b \)[/tex].
So, the equation [tex]\( q = 2p + \frac{1}{3} \)[/tex] can be interpreted in terms of the linear equation format as follows:
- Dependent variable ([tex]\( y \)[/tex] equivalent): [tex]\( q \)[/tex]
- Independent variable ([tex]\( x \)[/tex] equivalent): [tex]\( p \)[/tex]
- Slope ([tex]\( m \)[/tex]): [tex]\( 2 \)[/tex]
- Y-intercept ([tex]\( b \)[/tex]): [tex]\( \frac{1}{3} \)[/tex]
### 4. [tex]\( v = u \)[/tex]
The equation [tex]\( v = u \)[/tex] is a straightforward linear relationship where:
- [tex]\( v \)[/tex] and [tex]\( u \)[/tex] are variables that are equal to each other.
- There is no explicit slope or y-intercept in this equation since it's a direct equality.
- We can think of [tex]\( v = u \)[/tex] as a special case of the linear equation [tex]\( y = mx + b \)[/tex] where the slope [tex]\( m \)[/tex] is [tex]\( 1 \)[/tex] and the y-intercept [tex]\( b \)[/tex] is [tex]\( 0 \)[/tex].
Thus, the corresponding relationship is:
- Dependent variable ([tex]\( y \)[/tex] equivalent): [tex]\( v \)[/tex]
- Independent variable ([tex]\( x \)[/tex] equivalent): [tex]\( u \)[/tex]
- Slope ([tex]\( m \)[/tex]): [tex]\( 1 \)[/tex]
- Y-intercept ([tex]\( b \)[/tex]): [tex]\( 0 \)[/tex]
This indicates that for every unit increase in [tex]\( u \)[/tex], [tex]\( v \)[/tex] increases by the same amount. The line representing this relationship would be a 45-degree line passing through the origin on a graph with equal scales for [tex]\( u \)[/tex] and [tex]\( v \)[/tex].
In summary:
For the equation [tex]\( q = 2p + \frac{1}{3} \)[/tex]:
- Dependent variable: [tex]\( q \)[/tex]
- Independent variable: [tex]\( p \)[/tex]
- Slope [tex]\( m \)[/tex]: [tex]\( 2 \)[/tex]
- Y-intercept [tex]\( b \)[/tex]: [tex]\( \frac{1}{3} \)[/tex]
For the equation [tex]\( v = u \)[/tex]:
- Dependent variable: [tex]\( v \)[/tex]
- Independent variable: [tex]\( u \)[/tex]
- Slope [tex]\( m \)[/tex]: [tex]\( 1 \)[/tex]
- Y-intercept [tex]\( b \)[/tex]: [tex]\( 0 \)[/tex]
