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What is the equation of the line that passes through the points ( 3 , 10 ) (-3,-10) and ( 7 , 5 ) -7,5) ? Write your answer in slope-intercept form.
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To find the equation of the line that passes through the points ( 3 , 10 ) (-3,-10) and ( 7 , 5 ) (-7,5), we can start by finding the slope of the line. The slope of a line passing through two points ( x 1 , y 1 ) (x_(1),y_(1)) and ( x 2 , y 2 ) (x_(2),y_(2)) is given by the formula:
m = y 2 y 1 x 2 x 1 m=(y_(2)-y_(1))/(x_(2)-x_(1))
Substituting the coordinates of the two points, we get:
m = 5 ( 10 ) 7 ( 3 ) = 15 4 = 15 4 m=(5-(-10))/(-7-(-3))=(15)/(-4)=-(15)/(4)
So the slope of the line passing through the two points is 15 4 -(15)/(4).
Next, we need to find the y y-intercept of the line. To do this, we can use the point-slope form of a linear equation:
y y 1 = m ( x x 1 ) y-y_(1)=m(x-x_(1))
Choose either point to substitute as ( x 1 , y 1 ) (x_(1),y_(1)), and use the slope we just found. We’ll use the first point ( x 1 , y 1 ) = ( 3 , 10 ) (x_(1),y_(1))=(-3,-10):
y ( 10 ) = 15 4 ( x ( 3 ) ) y-(-10)=-(15)/(4)(x-(-3))
Simplifying this equation:
y + 10 = 15 4 ( x + 3 ) y+10=-(15)/(4)(x+3)
Multiplying both sides by 4:
4 y + 40 = 15 ( x + 3 ) 4y+40=-15(x+3)
Distributing the 15 -15:
4 y + 40 = 15 x 45 4y+40=-15 x-45
Solving for y y:
4 y = 15 x 45 40 = 15 x 85 4y=-15 x-45-40=-15 x-85
y = 15 4 x 85 4 y=-(15)/(4)x-(85)/(4)
So the equation of the line that passes through the points ( 3 , 10 ) (-3,-10) and ( 7 , 5 ) (-7,5) is:
y = 15 4 x 85 4 y=-(15)/(4)x-(85)/(4)
written in slope-intercept form.
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