Answer:
The equations represent the line that is parallel to 3x - 4y = 7 and pass through the point (-4,-2) are:
Step-by-step explanation:
The slope-intercept form of the line equation
[tex]y = mx+b[/tex]
where
Given the line
3x - 4y = 7
writing in the slope-intercept form
4y = 3x - 7
dividing both sides by 4
4y/4 = 3/4x - 7/4
y = 3/4x - 7/4
Now, comparing with the slope-intercept form of the line equation
y = 3/4x - 7/4
The slope of the line m = 3/4
We know that parallel lines have the same slopes.
Therefore, the slope of the parallel line is: 3/4
now we have,
The point (-4, -2)
The slope m of parallel line = 3/4
Given the point-slope form of the line equation
[tex]y-y_1=m\left(x-x_1\right)[/tex]
where m is the slope of the line and (x₁, y₁) is the point
substituting (-4, -2) and m = 3/4 in the point-slope form of line equation
[tex]\:y-\left(-2\right)=\frac{3}{4}\left(x-\left(-4\right)\right)[/tex]
[tex]y+2=\frac{3}{4}\left(x+4\right)[/tex]
Thus, the equation in the point-slope form of the line equation is:
[tex]y+2=\frac{3}{4}\left(x+4\right)[/tex]
Simplifying the equation
[tex]y+2=\frac{3}{4}\left(x+4\right)[/tex]
Subtract 3 from both sides
[tex]y+2-2=\frac{3}{4}\left(x+4\right)-2[/tex]
[tex]y=\frac{3}{4}x+1[/tex]
Multiplying the equation by 4
[tex]4y = 3x + 4[/tex]
[tex]3x - 4y = -4[/tex]
Therefore, the equations represent the line that is parallel to 3x - 4y = 7 and pass through the point (-4,-2) are:
The equations that represent the line that is parallel to 3x - 4y = 7 and passes through (-4, -2) are: y + 2 = 3/4x + 3 and y = 3/4x + 1
If two lines are parallel, the slope (m), in their equation in slope-intercept form (y = mx + b) would be the same.
Rewrite the equation, 3x - 4y = 7 in slope-intercept form
-4y = -3x + 7
y = -3y/-4 + 7/-4
y = 3/4y - 7/4
The slope (m) is 3/4, therefore, the slope of the line that is parallel to 3x - 4y = 7 and passes through (-4, -2) would be: m = 3/4.
Substitute (a, b) = (-4, -2) and m = 3/4 into y - b = m(x - a)
y -(-2) = 3/4(x - (-4))
y + 2 = 3/4(x + 4)
y + 2 = 3/4x + 3
y = 3/4x + 3 - 2
y = 3/4x + 1
The equations that represent the line are:
y + 2 = 3/4x + 3 and y = 3/4x + 1
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