Answer :
To find the equation of the line that passes through the points [tex]\((24, -3)\)[/tex] and [tex]\((12, -5)\)[/tex], we need to determine the slope and then use it to write the equation in different forms: Point-Slope Form, Slope-Intercept Form, and Standard Form.
### Finding the Slope
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points [tex]\((24, -3)\)[/tex] and [tex]\((12, -5)\)[/tex]:
[tex]\[ m = \frac{-5 - (-3)}{12 - 24} = \frac{-5 + 3}{-12} = \frac{-2}{-12} = \frac{1}{6} \approx 0.1667 \][/tex]
### Point-Slope Form
The Point-Slope Form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using the point [tex]\((24, -3)\)[/tex] and the slope [tex]\( m \approx 0.1667 \)[/tex]:
[tex]\[ y - (-3) = \frac{1}{6}(x - 24) \][/tex]
So, the Point-Slope Form is:
[tex]\[ y + 3 = \frac{1}{6}(x - 24) \][/tex]
Or, in its approximate form:
[tex]\[ y - (-3) = 0.1667(x - 24) \][/tex]
### Slope-Intercept Form
The Slope-Intercept Form of the equation of a line is:
[tex]\[ y = mx + b \][/tex]
To find the y-intercept [tex]\( b \)[/tex], we'll use one of our points, for example, [tex]\((24, -3)\)[/tex]:
[tex]\[ -3 = \frac{1}{6} \cdot 24 + b \quad \Rightarrow \quad -3 = 4 + b \quad \Rightarrow \quad b = -7 \][/tex]
Thus, the Slope-Intercept Form is:
[tex]\[ y = \frac{1}{6}x - 7 \][/tex]
Or, in its approximate form:
[tex]\[ y = 0.1667x - 7 \][/tex]
### Standard Form
The Standard Form of the equation of a line is:
[tex]\[ Ax + By = C \][/tex]
Starting with the Slope-Intercept Form [tex]\( y = \frac{1}{6}x - 7 \)[/tex], we rearrange to Standard Form:
[tex]\[ y = \frac{1}{6}x - 7 \quad \Rightarrow \quad \frac{1}{6}x - y = 7 \quad \Rightarrow \quad x - 6y = 42 \][/tex]
So, the Standard Form is:
[tex]\[ x - 6y = 42 \][/tex]
Or, using exact forms from earlier calculations:
[tex]\[ -\frac{1}{6}x + y = -7 \][/tex]
Multiplied by 6 to clear the fraction:
So the consistent final form is:
[tex]\[ -x + 6y = -42 \][/tex]
### Summary
1. Point-Slope Form: [tex]\(\boldsymbol{y + 3 = \frac{1}{6}(x - 24)}\)[/tex]
2. Slope-Intercept Form: [tex]\(\boldsymbol{y = \frac{1}{6}x - 7}\)[/tex]
3. Standard Form: [tex]\(\boldsymbol{x - 6y = 42}\)[/tex]
These are the equations in their respective forms for the line passing through the points [tex]\((24, -3)\)[/tex] and [tex]\((12, -5)\)[/tex].
To solve and graph inequalities was mentioned but no inequalities are provided. If you need additional help with that part, please provide the inequalities.
### Finding the Slope
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points [tex]\((24, -3)\)[/tex] and [tex]\((12, -5)\)[/tex]:
[tex]\[ m = \frac{-5 - (-3)}{12 - 24} = \frac{-5 + 3}{-12} = \frac{-2}{-12} = \frac{1}{6} \approx 0.1667 \][/tex]
### Point-Slope Form
The Point-Slope Form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using the point [tex]\((24, -3)\)[/tex] and the slope [tex]\( m \approx 0.1667 \)[/tex]:
[tex]\[ y - (-3) = \frac{1}{6}(x - 24) \][/tex]
So, the Point-Slope Form is:
[tex]\[ y + 3 = \frac{1}{6}(x - 24) \][/tex]
Or, in its approximate form:
[tex]\[ y - (-3) = 0.1667(x - 24) \][/tex]
### Slope-Intercept Form
The Slope-Intercept Form of the equation of a line is:
[tex]\[ y = mx + b \][/tex]
To find the y-intercept [tex]\( b \)[/tex], we'll use one of our points, for example, [tex]\((24, -3)\)[/tex]:
[tex]\[ -3 = \frac{1}{6} \cdot 24 + b \quad \Rightarrow \quad -3 = 4 + b \quad \Rightarrow \quad b = -7 \][/tex]
Thus, the Slope-Intercept Form is:
[tex]\[ y = \frac{1}{6}x - 7 \][/tex]
Or, in its approximate form:
[tex]\[ y = 0.1667x - 7 \][/tex]
### Standard Form
The Standard Form of the equation of a line is:
[tex]\[ Ax + By = C \][/tex]
Starting with the Slope-Intercept Form [tex]\( y = \frac{1}{6}x - 7 \)[/tex], we rearrange to Standard Form:
[tex]\[ y = \frac{1}{6}x - 7 \quad \Rightarrow \quad \frac{1}{6}x - y = 7 \quad \Rightarrow \quad x - 6y = 42 \][/tex]
So, the Standard Form is:
[tex]\[ x - 6y = 42 \][/tex]
Or, using exact forms from earlier calculations:
[tex]\[ -\frac{1}{6}x + y = -7 \][/tex]
Multiplied by 6 to clear the fraction:
So the consistent final form is:
[tex]\[ -x + 6y = -42 \][/tex]
### Summary
1. Point-Slope Form: [tex]\(\boldsymbol{y + 3 = \frac{1}{6}(x - 24)}\)[/tex]
2. Slope-Intercept Form: [tex]\(\boldsymbol{y = \frac{1}{6}x - 7}\)[/tex]
3. Standard Form: [tex]\(\boldsymbol{x - 6y = 42}\)[/tex]
These are the equations in their respective forms for the line passing through the points [tex]\((24, -3)\)[/tex] and [tex]\((12, -5)\)[/tex].
To solve and graph inequalities was mentioned but no inequalities are provided. If you need additional help with that part, please provide the inequalities.
