# Find the equation of the line that passes through (3,4) and the sum of it's intercepts on the axis is 14.

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The equation for the line is:

y-y1 = m (x-x1)

We have the point (3,4) psses through the line,

==> y-4 = m(x-3)

==> y = mx - 3m + 4

==> y - mx = -3m + 4

Divide by -3m + 4:

==> y/(-3m+4) + x/[(-3m+4)/-m] = 1

==> y intercept a = -3m+4

==> x intercept b = (-3m+4)/-m

But we know that a+ b = 14

==> -3m + 4 + (-3m+4)/-m = 14

==> -3m + 4 + 3 - 4/m = 14

==> -3m^2 + 7m - 4 = 14m

==> -3m^2 - 7m - 4 = 0

==> m1= [7+ sqrt(49-48)]/-6 = [7+1]/-6 = -8/6= -4/3

==> m2= [7-1]/-6= 6/-6 = -1

==> we have two solutions:

m= -4/3:

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

==> y= (-4/3)x + 8

m= -1:

==> y-4 = (-1)(x-3)

==> y= -x +7

Let a and b be the intercepts of a line on x and y axis.

Then a+b =14.

So b= 14-a.

Then the equation of the line is x/a+y/b = 1, double intercept form of the line. Or

x/a+y/(14-b) = 1 . This passes through (3,4).

So 3/a +4/(14-a) = 1

3(14-a)+4a = a(14-a)

42-3a+4a = 14a-a^2

42+a =14a-a^2

a^2 - 14a+a+42 = 0

a^2-13a+42 = 0

(a-6)(a-7) = 0

a = 6 or a =7.

Therefore the b = 14-a = 8 when a = 6. Or b = 14-7 = 7 when a = 7.

So the equation of the required lie is x/6+y/8 =1 or Multiplying by 24, we get 4x+3y = 24, when a = 6 and b = 8.

x/7+y/7 =1 or x+y = 7. when a = 7, b= 7.

Or

The intercept form of the equation of the line is:

x/a + y/b = 1,

where a is x intercept and b is y intercept.

We know, from enunciation that the sum of intercept is 14.

a+b = 14 => b = 14 - a

We'll re-write the equation:

x/a + y/(14 - a) = 1

We know that the line passes through the point (3,4).

We'll input the coordinates of the point into the equation of the line:

3/a + 4/(14-a) = 1

We'll calculate LCD:

3(14-a) + 4a = a(14-a)

We'll remove the brackets:

42 - 3a + 4a = 14a - a^2

42 + a - 14a + a^2 = 0

a^2 - 13a + 42 = 0

We'll pply the quadratic formula:

a1 = [13+sqrt(169-168)]/2

a1 = (13+1)/2

a1 = 7

a2 = (13-1)/2

a2 = 6

We'll write the equations for both values of a:

For a = 7 => x/7 + y/(14 - 7) = 1

x/7 + y/7 = 1

**x + y - 7 = 0**

For a = 6 => x/6 + y/(14 - 6) = 1

x/6 + y/8 = 1

**4x + 3y - 24 = 0**