Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Experience the ease of finding reliable answers to your questions from a vast community of knowledgeable experts. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Certainly! Let's go through the process of finding the equation of the line perpendicular to [tex]\(WX\)[/tex] and passing through the point [tex]\((-1, -2)\)[/tex] step-by-step.
### Step 1: Convert the Equation of [tex]\(WX\)[/tex] to Slope-Intercept Form
The given equation of line [tex]\(WX\)[/tex] is:
[tex]\[ 2x + y = -5 \][/tex]
To convert this to slope-intercept form ([tex]\(y = mx + b\)[/tex]), we need to solve for [tex]\(y\)[/tex]:
[tex]\[ y = -2x - 5 \][/tex]
Therefore, the slope ([tex]\(m\)[/tex]) of line [tex]\(WX\)[/tex] is [tex]\(-2\)[/tex].
### Step 2: Find the Slope of the Perpendicular Line
The slope of the line perpendicular to another is the negative reciprocal of the original slope. Thus, the slope of the line perpendicular to [tex]\(WX\)[/tex] is:
[tex]\[ \text{Slope}_{\text{perpendicular}} = -\frac{1}{\text{Slope}_{\text{WX}}} = -\frac{1}{-2} = \frac{1}{2} \][/tex]
### Step 3: Use the Point-Slope Form to Find the Equation
We now know the slope of the perpendicular line is [tex]\(\frac{1}{2}\)[/tex], and it passes through the point [tex]\((-1, -2)\)[/tex]. We can use the point-slope form of the equation of a line ([tex]\(y - y_1 = m (x - x_1)\)[/tex]) to find the equation.
Given:
[tex]\[ m = \frac{1}{2}, \quad (x_1, y_1) = (-1, -2) \][/tex]
Substitute these values into the point-slope form:
[tex]\[ y - (-2) = \frac{1}{2}(x - (-1)) \][/tex]
Simplify:
[tex]\[ y + 2 = \frac{1}{2} (x + 1) \][/tex]
[tex]\[ y + 2 = \frac{1}{2}x + \frac{1}{2} \][/tex]
### Step 4: Convert to Slope-Intercept Form
To convert this into slope-intercept form ([tex]\(y = mx + b\)[/tex]), we subtract 2 from both sides:
[tex]\[ y = \frac{1}{2}x + \frac{1}{2} - 2 \][/tex]
[tex]\[ y = \frac{1}{2}x - \frac{3}{2} \][/tex]
### Step 5: Select the Correct Answer
Thus, the equation of the line perpendicular to [tex]\(WX\)[/tex] and passing through the point [tex]\((-1, -2)\)[/tex] in slope-intercept form is:
[tex]\[ y = \frac{1}{2}x - \frac{3}{2} \][/tex]
So, the correct answer is:
[tex]\[ y = \frac{1}{2}x - \frac{3}{2} \][/tex]
This corresponds to the third option:
[tex]\[ \boxed{y = \frac{1}{2} x - \frac{3}{2}} \][/tex]
### Step 1: Convert the Equation of [tex]\(WX\)[/tex] to Slope-Intercept Form
The given equation of line [tex]\(WX\)[/tex] is:
[tex]\[ 2x + y = -5 \][/tex]
To convert this to slope-intercept form ([tex]\(y = mx + b\)[/tex]), we need to solve for [tex]\(y\)[/tex]:
[tex]\[ y = -2x - 5 \][/tex]
Therefore, the slope ([tex]\(m\)[/tex]) of line [tex]\(WX\)[/tex] is [tex]\(-2\)[/tex].
### Step 2: Find the Slope of the Perpendicular Line
The slope of the line perpendicular to another is the negative reciprocal of the original slope. Thus, the slope of the line perpendicular to [tex]\(WX\)[/tex] is:
[tex]\[ \text{Slope}_{\text{perpendicular}} = -\frac{1}{\text{Slope}_{\text{WX}}} = -\frac{1}{-2} = \frac{1}{2} \][/tex]
### Step 3: Use the Point-Slope Form to Find the Equation
We now know the slope of the perpendicular line is [tex]\(\frac{1}{2}\)[/tex], and it passes through the point [tex]\((-1, -2)\)[/tex]. We can use the point-slope form of the equation of a line ([tex]\(y - y_1 = m (x - x_1)\)[/tex]) to find the equation.
Given:
[tex]\[ m = \frac{1}{2}, \quad (x_1, y_1) = (-1, -2) \][/tex]
Substitute these values into the point-slope form:
[tex]\[ y - (-2) = \frac{1}{2}(x - (-1)) \][/tex]
Simplify:
[tex]\[ y + 2 = \frac{1}{2} (x + 1) \][/tex]
[tex]\[ y + 2 = \frac{1}{2}x + \frac{1}{2} \][/tex]
### Step 4: Convert to Slope-Intercept Form
To convert this into slope-intercept form ([tex]\(y = mx + b\)[/tex]), we subtract 2 from both sides:
[tex]\[ y = \frac{1}{2}x + \frac{1}{2} - 2 \][/tex]
[tex]\[ y = \frac{1}{2}x - \frac{3}{2} \][/tex]
### Step 5: Select the Correct Answer
Thus, the equation of the line perpendicular to [tex]\(WX\)[/tex] and passing through the point [tex]\((-1, -2)\)[/tex] in slope-intercept form is:
[tex]\[ y = \frac{1}{2}x - \frac{3}{2} \][/tex]
So, the correct answer is:
[tex]\[ y = \frac{1}{2}x - \frac{3}{2} \][/tex]
This corresponds to the third option:
[tex]\[ \boxed{y = \frac{1}{2} x - \frac{3}{2}} \][/tex]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
