Question
Mathematics
What is the equation of the line that passes through the points $\left(-3,-10\right)$(-3,-10) and ( $-7,5\right)$-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 $\left(-3,-10\right)$(-3,-10) and $\left(-7,5\right)$(-7,5), we can start by finding the slope of the line. The slope of a line passing through two points $\left({x}_{1},{y}_{1}\right)$(x_(1),y_(1)) and $\left({x}_{2},{y}_{2}\right)$(x_(2),y_(2)) is given by the formula:
$m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$m=(y_(2)-y_(1))/(x_(2)-x_(1))
Substituting the coordinates of the two points, we get:
$m=\frac{5-\left(-10\right)}{-7-\left(-3\right)}=\frac{15}{-4}=-\frac{15}{4}$m=(5-(-10))/(-7-(-3))=(15)/(-4)=-(15)/(4)
So the slope of the line passing through the two points is $-\frac{15}{4}$-(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:
$y-{y}_{1}=m\left(x-{x}_{1}\right)$y-y_(1)=m(x-x_(1))
Choose either point to substitute as $\left({x}_{1},{y}_{1}\right)$(x_(1),y_(1)), and use the slope we just found. We’ll use the first point $\left({x}_{1},{y}_{1}\right)=\left(-3,-10\right)$(x_(1),y_(1))=(-3,-10):
$y-\left(-10\right)=-\frac{15}{4}\left(x-\left(-3\right)\right)$y-(-10)=-(15)/(4)(x-(-3))
Simplifying this equation:
$y+10=-\frac{15}{4}\left(x+3\right)$y+10=-(15)/(4)(x+3)
Multiplying both sides by 4:
$4y+40=-15\left(x+3\right)$4y+40=-15(x+3)
Distributing the $-15$-15:
$4y+40=-15x-45$4y+40=-15 x-45
Solving for $y$y:
$4y=-15x-45-40=-15x-85$4y=-15 x-45-40=-15 x-85
$y=-\frac{15}{4}x-\frac{85}{4}$y=-(15)/(4)x-(85)/(4)
So the equation of the line that passes through the points $\left(-3,-10\right)$(-3,-10) and $\left(-7,5\right)$(-7,5) is:
$y=-\frac{15}{4}x-\frac{85}{4}$y=-(15)/(4)x-(85)/(4)
written in slope-intercept form.
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