Find the equation of the line which passes through the point (3,4)and the sum of is intersept on the axes is 14.

Question

We have to find the equation of the line which is passing through point (3,4)

Answer 1

The following part of your problem is expressed quite poorly:

...the sum of is intersept on the axes is 14.

 

Granted, your line contains point (3,4).  If the x-intercept is 14, i.e. point (14,0) is also on the line, then you have two points and you can calculate the slope from this information.  m = (4-0)/(3-14) = -4/11.

 

Now you can find the equation using Point Slope Form. Are you able to complete this work?

Answer 2

The three points for reference are (0, y) , (x, 0), and (3, 4).  Since we have a line the slope between consecutive points are the same.  In other words, the slope between the 1st and 2nd point is equal to the slope between the 2nd and 3rd point.  It is important to use all points on this line for accuracy.

 

(0 - y) / (x - 0) = (4 - 0) / (3 - x)

 

-y / x = 4 / (3 - x)

 

 

Cross-multiply to get

 

4x = -y(3 - x)

4x = -3y + xy                      eq1

 

 

Then we use the equation for the sum of the intercepts.

 

x + y = 14                       eq2

 

 

Solve the system to solve for x and y.  This will give you the x-intercept and y-intercept.

 

4x = -3y + xy                      eq1

x + y = 14                           eq2

 

Substitute eq2 into eq1 to get eq1 in terms of x.

 

4x = -3(14 - x) + x(14 - x)

4x = -42 + 3x + 14x - x²

4x = -42 + 17x - x²

 

x² - 13x + 42 = 0

 

(x - 7)(x - 6) = 0

 

x = 6          or        x = 7

 

 

Plugging this values of x into eq2 gives

 

y = 8          or        y = 7

 

 

Plugging in x=7 and y=7 into eq1 does not satisfy.  Lets check the pair x=6 and y=8.

 

-8 / 6 = 4 / (3 - 6)

 

-4 / 3 = 4 / - 3

 

 This satisfies.  Thus, our slope is -4/3.

 

Now use the point-slope form of a line.

 

y - y₁ = m(x - x₁)

 

where:

m = -4/3

(x₁ , y₁) = (3, 4)