d1:x+y = 8

d2:4x - 2y = 12

To find if both line intersecting we will rewrite using the slope form.

d1: y= -x + 8

d2: y= 2x - 6

From the first look we notice that the slopes are not equal.

(m1= -1 , m2= 2) then the lines are NOT parallel, then they intersect.

To determine the intercepting point, then we need to find x and y values such that d1 = d2

==> -x +8 = 2x -6

==> -3x = -14

==> x= 14/3

Now to find y, we will substitute with either equation.

y= -x + 8

==> y= -14/3 + 8 = 10/3

**Then both lines intersect at the point (14/3 , 10/3)**

If the given lines are intercepting, then the system of equations has a solution. The solution of the system represents the coordinates of the intercepting point.

We'll solve the system using the elimination method.

We'll note the equations of the system as:

x+y = 8 (1)

4x-2y = 12 (2)

We'll multiply (1) by 2:

2x+2y = 16 (3)

We'll add (3) to (2):

2x+2y+4x-2y=16+12

We'll eliminate like terms:

6x = 28

We'll divide by 2:

3x = 14

We'll divide by 3:

**x = 14/3**

We'll substitute x in (1):

14/3 + y = 8 (1)

We'll subtract 14/3 both sides:

y = 8 - 14/3

y = (24-14)/3

**y = 10/3**

**The solution of the system represents the coordinates of the intercepting point {(14/3 ; 10/3)}.**

To verify whether the lones are intercepting.

x+y = 8

4x-2y = 12.

Two lines are intersecting if they are not parallel.

The given line are not parallel as the ratio of coefficiet x and y of 1st line is not that of the other line. Or their slope are different.

Let us solve the equations and the solution in x and y coordinates are the point of intersection of the 2 lines.

x+y = 8 an

4x-2y = 12.

Eq(1)*2 + eq(2) elimintes y:

(x+y)*2 +(4x-2y) = 8*2+12 = 28

6x = 28. x = 28/6 = 14/3.

So from (1) x+y= 8. Or y = 8-x = 8-14/3 = 10/3.

So (x,y) = (14/3 , 10/3) is the point of intersection of the lines.

Or

We can right the two lines in the form y= mx + b where m is the slope of the line and b is the y-intercept.

x+y = 8

=> y = -x + 8

4x - 2y =12

=> -2y = 12- 4x

=> y = -4x / -2 +12 / -2

=> y = 2x - 6

Therefore as the slopes of the two lines are 2 and -1 they intersect.

Now to see where they intersect, equate the values of y,

2x -6 = -x +8

=> 3x = 8+6 =14

=> x =14 / 3

Substituting this in y= -x + 8 = - 14/3 +8 = 10/3

**So the point of intersection is ( 14/3 , 10/3)**