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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]
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