Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To determine which equations represent a line that is parallel to [tex]\(3x - 4y = 7\)[/tex] and passes through the point [tex]\((-4, -2)\)[/tex], follow these detailed steps:
### Step 1: Find the slope of the given line
The given line is [tex]\(3x - 4y = 7\)[/tex]. To find the slope, we first rearrange this into the slope-intercept form [tex]\(y = mx + b\)[/tex].
Starting with:
[tex]\[ 3x - 4y = 7 \][/tex]
Rearrange to isolate [tex]\(y\)[/tex]:
[tex]\[ -4y = -3x + 7 \][/tex]
Divide by [tex]\(-4\)[/tex]:
[tex]\[ y = \frac{3}{4}x - \frac{7}{4} \][/tex]
The slope [tex]\(m\)[/tex] of the line is [tex]\(\frac{3}{4}\)[/tex].
### Step 2: Identify the form of the equation for a parallel line
A line parallel to the given line must have the same slope. Therefore, the slope of our line is also [tex]\(\frac{3}{4}\)[/tex].
### Step 3: Write the equation of the line passing through the point [tex]\((-4, -2)\)[/tex]
We use the point-slope form of a linear equation, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\(m = \frac{3}{4}\)[/tex], [tex]\((x_1, y_1) = (-4, -2)\)[/tex]. Substituting these values in:
[tex]\[ y - (-2) = \frac{3}{4}(x - (-4)) \][/tex]
Simplify:
[tex]\[ y + 2 = \frac{3}{4}(x + 4) \][/tex]
### Step 4: Examine the given options
Now, look at the provided options to see which ones match the criteria of having the same slope and passing through [tex]\((-4, -2)\)[/tex].
Option 1: [tex]\( y = \frac{3}{4}x + 1 \)[/tex]
This equation is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m = \frac{3}{4}\)[/tex]. It has the correct slope. Let's check if it passes through [tex]\((-4, -2)\)[/tex].
Substitute [tex]\((-4, -2)\)[/tex]:
[tex]\[ -2 = \frac{3}{4}(-4) + 1 \][/tex]
[tex]\[ -2 = -3 + 1 \][/tex]
[tex]\[ -2 = -2 \][/tex]
It passes through [tex]\((-4, -2)\)[/tex].
Option 2: [tex]\( 3x - 4y = -4 \)[/tex]
Rewrite in slope-intercept form:
[tex]\[ 3x - 4y = -4 \][/tex]
[tex]\[ -4y = -3x - 4 \][/tex]
[tex]\[ y = \frac{3}{4}x + 1 \][/tex]
This slope is [tex]\(\frac{3}{4}\)[/tex]. Check if it passes through [tex]\((-4, -2)\)[/tex]:
[tex]\[ -2 = \frac{3}{4}(-4) + 1 \][/tex]
[tex]\[ -2 = -3 + 1 \][/tex]
[tex]\[ -2 = -2 \][/tex]
It passes through [tex]\((-4, -2)\)[/tex].
Option 3: [tex]\( 4x - 3y = -3 \)[/tex]
Rearrange to find the slope:
[tex]\[ -3y = -4x - 3 \][/tex]
[tex]\[ y = \frac{4}{3}x + 1 \][/tex]
The slope is [tex]\(\frac{4}{3}\)[/tex], which is not [tex]\(\frac{3}{4}\)[/tex], so this line is not parallel.
Option 4: [tex]\( y - 2 = \frac{3}{4}(x - 4) \)[/tex]
This is already in point-slope form, but for a point [tex]\((4, 2)\)[/tex], not [tex]\((-4, -2)\)[/tex]. Check for slope and point:
Option 5: [tex]\( y + 2 = \frac{3}{4}(x + 4) \)[/tex]
This is the equation we derived in Step 3. It has the correct slope and passes through [tex]\((-4, -2)\)[/tex].
### Conclusion
The equations that represent the line parallel to [tex]\(3x - 4y = 7\)[/tex] and passing through [tex]\((-4, -2)\)[/tex] are:
1. [tex]\( y = \frac{3}{4}x + 1 \)[/tex]
2. [tex]\( y + 2 = \frac{3}{4}(x + 4) \)[/tex]
Thus, the correct options are:
[tex]\[ \boxed{y = \frac{3}{4} x + 1 \text{ and } y + 2 = \frac{3}{4}(x + 4)} \][/tex]
### Step 1: Find the slope of the given line
The given line is [tex]\(3x - 4y = 7\)[/tex]. To find the slope, we first rearrange this into the slope-intercept form [tex]\(y = mx + b\)[/tex].
Starting with:
[tex]\[ 3x - 4y = 7 \][/tex]
Rearrange to isolate [tex]\(y\)[/tex]:
[tex]\[ -4y = -3x + 7 \][/tex]
Divide by [tex]\(-4\)[/tex]:
[tex]\[ y = \frac{3}{4}x - \frac{7}{4} \][/tex]
The slope [tex]\(m\)[/tex] of the line is [tex]\(\frac{3}{4}\)[/tex].
### Step 2: Identify the form of the equation for a parallel line
A line parallel to the given line must have the same slope. Therefore, the slope of our line is also [tex]\(\frac{3}{4}\)[/tex].
### Step 3: Write the equation of the line passing through the point [tex]\((-4, -2)\)[/tex]
We use the point-slope form of a linear equation, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\(m = \frac{3}{4}\)[/tex], [tex]\((x_1, y_1) = (-4, -2)\)[/tex]. Substituting these values in:
[tex]\[ y - (-2) = \frac{3}{4}(x - (-4)) \][/tex]
Simplify:
[tex]\[ y + 2 = \frac{3}{4}(x + 4) \][/tex]
### Step 4: Examine the given options
Now, look at the provided options to see which ones match the criteria of having the same slope and passing through [tex]\((-4, -2)\)[/tex].
Option 1: [tex]\( y = \frac{3}{4}x + 1 \)[/tex]
This equation is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m = \frac{3}{4}\)[/tex]. It has the correct slope. Let's check if it passes through [tex]\((-4, -2)\)[/tex].
Substitute [tex]\((-4, -2)\)[/tex]:
[tex]\[ -2 = \frac{3}{4}(-4) + 1 \][/tex]
[tex]\[ -2 = -3 + 1 \][/tex]
[tex]\[ -2 = -2 \][/tex]
It passes through [tex]\((-4, -2)\)[/tex].
Option 2: [tex]\( 3x - 4y = -4 \)[/tex]
Rewrite in slope-intercept form:
[tex]\[ 3x - 4y = -4 \][/tex]
[tex]\[ -4y = -3x - 4 \][/tex]
[tex]\[ y = \frac{3}{4}x + 1 \][/tex]
This slope is [tex]\(\frac{3}{4}\)[/tex]. Check if it passes through [tex]\((-4, -2)\)[/tex]:
[tex]\[ -2 = \frac{3}{4}(-4) + 1 \][/tex]
[tex]\[ -2 = -3 + 1 \][/tex]
[tex]\[ -2 = -2 \][/tex]
It passes through [tex]\((-4, -2)\)[/tex].
Option 3: [tex]\( 4x - 3y = -3 \)[/tex]
Rearrange to find the slope:
[tex]\[ -3y = -4x - 3 \][/tex]
[tex]\[ y = \frac{4}{3}x + 1 \][/tex]
The slope is [tex]\(\frac{4}{3}\)[/tex], which is not [tex]\(\frac{3}{4}\)[/tex], so this line is not parallel.
Option 4: [tex]\( y - 2 = \frac{3}{4}(x - 4) \)[/tex]
This is already in point-slope form, but for a point [tex]\((4, 2)\)[/tex], not [tex]\((-4, -2)\)[/tex]. Check for slope and point:
Option 5: [tex]\( y + 2 = \frac{3}{4}(x + 4) \)[/tex]
This is the equation we derived in Step 3. It has the correct slope and passes through [tex]\((-4, -2)\)[/tex].
### Conclusion
The equations that represent the line parallel to [tex]\(3x - 4y = 7\)[/tex] and passing through [tex]\((-4, -2)\)[/tex] are:
1. [tex]\( y = \frac{3}{4}x + 1 \)[/tex]
2. [tex]\( y + 2 = \frac{3}{4}(x + 4) \)[/tex]
Thus, the correct options are:
[tex]\[ \boxed{y = \frac{3}{4} x + 1 \text{ and } y + 2 = \frac{3}{4}(x + 4)} \][/tex]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
