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Sagot :
Let's start by determining the slope of the given line [tex]\(3x - 4y = 7\)[/tex]. To find the slope, we'll rewrite this equation in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
First, solve for [tex]\(y\)[/tex]:
[tex]\[ 3x - 4y = 7 \][/tex]
Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[ -4y = -3x + 7 \][/tex]
Divide every term by [tex]\(-4\)[/tex]:
[tex]\[ y = \frac{3}{4}x - \frac{7}{4} \][/tex]
The slope [tex]\(m\)[/tex] of the given line is [tex]\(\frac{3}{4}\)[/tex].
Any line parallel to this one will have the same slope, [tex]\(\frac{3}{4}\)[/tex]. Now, let's examine each of the given options to see if they both have this slope and go through the point [tex]\((-4, -2)\)[/tex]:
1. Option 1: [tex]\(y = -\frac{3}{4}x + 1\)[/tex]
The slope of this line is [tex]\(-\frac{3}{4}\)[/tex], which is different from [tex]\(\frac{3}{4}\)[/tex]. Thus, this line is not parallel to the given line.
2. Option 2: [tex]\(3x - 4y = -4\)[/tex]
To find the slope, we can rearrange this equation into slope-intercept form:
[tex]\[ 3x - 4y = -4 \][/tex]
Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[ -4y = -3x - 4 \][/tex]
Divide every term by [tex]\(-4\)[/tex]:
[tex]\[ y = \frac{3}{4}x + 1 \][/tex]
The slope is [tex]\(\frac{3}{4}\)[/tex], which matches the slope of the given line. Now we need to check if it passes through the point [tex]\((-4, -2)\)[/tex]. Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = -2\)[/tex] into the equation [tex]\(y = \frac{3}{4}x + 1\)[/tex]:
[tex]\[ -2 = \frac{3}{4}(-4) + 1 \][/tex]
Simplify:
[tex]\[ -2 = -3 + 1 \][/tex]
[tex]\[ -2 = -2 \][/tex]
This is true, so Option 2 is correct.
3. Option 3: [tex]\(4x - 3y = -3\)[/tex]
To find the slope, we can rearrange this equation into slope-intercept form:
[tex]\[ 4x - 3y = -3 \][/tex]
Subtract [tex]\(4x\)[/tex] from both sides:
[tex]\[ -3y = -4x - 3 \][/tex]
Divide every term by [tex]\(-3\)[/tex]:
[tex]\[ y = \frac{4}{3}x + 1 \][/tex]
The slope is [tex]\(\frac{4}{3}\)[/tex], which is different from [tex]\(\frac{3}{4}\)[/tex]. Thus, this line is not parallel to the given line.
4. Option 4: [tex]\(y - 2 = -\frac{3}{4}(x - 4)\)[/tex]
First, convert this into slope-intercept form:
Distribute [tex]\(-\frac{3}{4}\)[/tex]:
[tex]\[ y - 2 = -\frac{3}{4}x + 3 \][/tex]
Add 2 to both sides:
[tex]\[ y = -\frac{3}{4}x + 5 \][/tex]
The slope is [tex]\(-\frac{3}{4}\)[/tex], which is different from [tex]\(\frac{3}{4}\)[/tex]. Thus, this line is not parallel to the given line.
5. Option 5: [tex]\(y + 2 = \frac{3}{4}(x + 4)\)[/tex]
First, convert this into slope-intercept form:
Distribute [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[ y + 2 = \frac{3}{4}x + 3 \][/tex]
Subtract 2 from both sides:
[tex]\[ y = \frac{3}{4}x + 1 \][/tex]
The slope is [tex]\(\frac{3}{4}\)[/tex], which matches the slope of the given line. Now we need to check if it passes through the point [tex]\((-4, -2)\)[/tex]. Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = -2\)[/tex] into the equation [tex]\(y = \frac{3}{4}x + 1\)[/tex]:
[tex]\[ -2 = \frac{3}{4}(-4) + 1 \][/tex]
Simplify:
[tex]\[ -2 = -3 + 1 \][/tex]
[tex]\[ -2 = -2 \][/tex]
This is true, so Option 5 is correct.
Thus, the equations that represent the line parallel to [tex]\(3x - 4y = 7\)[/tex] and pass through the point [tex]\((-4, -2)\)[/tex] are Options 2 and 5.
First, solve for [tex]\(y\)[/tex]:
[tex]\[ 3x - 4y = 7 \][/tex]
Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[ -4y = -3x + 7 \][/tex]
Divide every term by [tex]\(-4\)[/tex]:
[tex]\[ y = \frac{3}{4}x - \frac{7}{4} \][/tex]
The slope [tex]\(m\)[/tex] of the given line is [tex]\(\frac{3}{4}\)[/tex].
Any line parallel to this one will have the same slope, [tex]\(\frac{3}{4}\)[/tex]. Now, let's examine each of the given options to see if they both have this slope and go through the point [tex]\((-4, -2)\)[/tex]:
1. Option 1: [tex]\(y = -\frac{3}{4}x + 1\)[/tex]
The slope of this line is [tex]\(-\frac{3}{4}\)[/tex], which is different from [tex]\(\frac{3}{4}\)[/tex]. Thus, this line is not parallel to the given line.
2. Option 2: [tex]\(3x - 4y = -4\)[/tex]
To find the slope, we can rearrange this equation into slope-intercept form:
[tex]\[ 3x - 4y = -4 \][/tex]
Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[ -4y = -3x - 4 \][/tex]
Divide every term by [tex]\(-4\)[/tex]:
[tex]\[ y = \frac{3}{4}x + 1 \][/tex]
The slope is [tex]\(\frac{3}{4}\)[/tex], which matches the slope of the given line. Now we need to check if it passes through the point [tex]\((-4, -2)\)[/tex]. Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = -2\)[/tex] into the equation [tex]\(y = \frac{3}{4}x + 1\)[/tex]:
[tex]\[ -2 = \frac{3}{4}(-4) + 1 \][/tex]
Simplify:
[tex]\[ -2 = -3 + 1 \][/tex]
[tex]\[ -2 = -2 \][/tex]
This is true, so Option 2 is correct.
3. Option 3: [tex]\(4x - 3y = -3\)[/tex]
To find the slope, we can rearrange this equation into slope-intercept form:
[tex]\[ 4x - 3y = -3 \][/tex]
Subtract [tex]\(4x\)[/tex] from both sides:
[tex]\[ -3y = -4x - 3 \][/tex]
Divide every term by [tex]\(-3\)[/tex]:
[tex]\[ y = \frac{4}{3}x + 1 \][/tex]
The slope is [tex]\(\frac{4}{3}\)[/tex], which is different from [tex]\(\frac{3}{4}\)[/tex]. Thus, this line is not parallel to the given line.
4. Option 4: [tex]\(y - 2 = -\frac{3}{4}(x - 4)\)[/tex]
First, convert this into slope-intercept form:
Distribute [tex]\(-\frac{3}{4}\)[/tex]:
[tex]\[ y - 2 = -\frac{3}{4}x + 3 \][/tex]
Add 2 to both sides:
[tex]\[ y = -\frac{3}{4}x + 5 \][/tex]
The slope is [tex]\(-\frac{3}{4}\)[/tex], which is different from [tex]\(\frac{3}{4}\)[/tex]. Thus, this line is not parallel to the given line.
5. Option 5: [tex]\(y + 2 = \frac{3}{4}(x + 4)\)[/tex]
First, convert this into slope-intercept form:
Distribute [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[ y + 2 = \frac{3}{4}x + 3 \][/tex]
Subtract 2 from both sides:
[tex]\[ y = \frac{3}{4}x + 1 \][/tex]
The slope is [tex]\(\frac{3}{4}\)[/tex], which matches the slope of the given line. Now we need to check if it passes through the point [tex]\((-4, -2)\)[/tex]. Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = -2\)[/tex] into the equation [tex]\(y = \frac{3}{4}x + 1\)[/tex]:
[tex]\[ -2 = \frac{3}{4}(-4) + 1 \][/tex]
Simplify:
[tex]\[ -2 = -3 + 1 \][/tex]
[tex]\[ -2 = -2 \][/tex]
This is true, so Option 5 is correct.
Thus, the equations that represent the line parallel to [tex]\(3x - 4y = 7\)[/tex] and pass through the point [tex]\((-4, -2)\)[/tex] are Options 2 and 5.
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