SOLUTION: Please help me understand how to writh an equation of the line through the points (4,-5) and parallel to the line 2x-3y =4 in standard form. Thank you

Algebra ->  Systems-of-equations -> SOLUTION: Please help me understand how to writh an equation of the line through the points (4,-5) and parallel to the line 2x-3y =4 in standard form. Thank you      Log On


   

Question 115237: Please help me understand how to writh an equation of the line through the points (4,-5) and parallel to the line 2x-3y =4 in standard form.
Thank you
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First convert 2x-3y =4 into slope-intercept form

Solved by pluggable solver: Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa)
Convert from standard form (Ax+By = C) to slope-intercept form (y = mx+b)


2x-3y=4 Start with the given equation


2x-3y-2x=4-2x Subtract 2x from both sides


-3y=-2x%2B4 Simplify


%28-3y%29%2F%28-3%29=%28-2x%2B4%29%2F%28-3%29 Divide both sides by -3 to isolate y


y+=+%28-2x%29%2F%28-3%29%2B%284%29%2F%28-3%29 Break up the fraction on the right hand side


y+=+%282%2F3%29x-4%2F3 Reduce and simplify


The original equation 2x-3y=4 (standard form) is equivalent to y+=+%282%2F3%29x-4%2F3 (slope-intercept form)


The equation y+=+%282%2F3%29x-4%2F3 is in the form y=mx%2Bb where m=2%2F3 is the slope and b=-4%2F3 is the y intercept.





Now let's find the line that is parallel to y=%282%2F3%29x-4%2F3 that goes through the point (4,-5)


Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line


Since any two parallel lines have the same slope we know the slope of the unknown line is 2%2F3 (its from the slope of y=%282%2F3%29%2Ax-4%2F3 which is also 2%2F3). Also since the unknown line goes through (4,-5), we can find the equation by plugging in this info into the point-slope formula

Point-Slope Formula:

y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope and (x%5B1%5D,y%5B1%5D) is the given point



y%2B5=%282%2F3%29%2A%28x-4%29 Plug in m=2%2F3, x%5B1%5D=4, and y%5B1%5D=-5



y%2B5=%282%2F3%29%2Ax-%282%2F3%29%284%29 Distribute 2%2F3



y%2B5=%282%2F3%29%2Ax-8%2F3 Multiply



y=%282%2F3%29%2Ax-8%2F3-5Subtract -5 from both sides to isolate y

y=%282%2F3%29%2Ax-8%2F3-15%2F3 Make into equivalent fractions with equal denominators



y=%282%2F3%29%2Ax-23%2F3 Combine the fractions



y=%282%2F3%29%2Ax-23%2F3 Reduce any fractions

So the equation of the line that is parallel to y=%282%2F3%29%2Ax-4%2F3 and goes through (4,-5) is y=%282%2F3%29%2Ax-23%2F3


So here are the graphs of the equations y=%282%2F3%29%2Ax-4%2F3 and y=%282%2F3%29%2Ax-23%2F3



graph of the given equation y=%282%2F3%29%2Ax-4%2F3 (red) and graph of the line y=%282%2F3%29%2Ax-23%2F3(green) that is parallel to the given graph and goes through (4,-5)