How do you find the point of intersection for #3x-y=4# and #6x+2y= -8#?

Question

Savannah Anderson · Accepted Answer

Intersection point : (0,-4)

We want to find the point #A(X,Y)# like :
#3X-Y=4# and #6X+2Y=-8#

The word "intersection", here, is referring to functions :
A function is generally writing : #y=f(x)#

Then, we need to transform the two equations to something like :
"#y=...#"
Let's define functions #f,g#, who are respectively representing equations #3x-y=4# and #6x+2y=-8#

Function #f# :
#3x - y = 4 <=> 3x = 4 + y <=> 3x-4 = y#
Then we have #f(x)=3x-4#

Function #g# :
#6x + 2y = -8 <=> 2y = -8 - 6x <=> y= -4-3x#
Then we have #g(x)=-3x-4#

#A(X,Y)# is an intersection point between #f# and #g# then :
#f(X) = Y# and #g(X)=Y#
We can mark here #f(X) = g(X)# and more :

#3X-4 = -3X-4#
#<=> 3X = -3X# (we added 4 to each side)
#<=> 6X = 0#
#<=> X = 0#

Then : #A(0,Y)# and #Y=f(0)=g(0)=-4#

The coordinates of #A# is #A(0,-4)#

We can check the result with a graph of the situation (Alone, this is not a proof !!)

Avery Alexander · Answer

undefined To find the point of intersection for the equations 3x - y = 4 and 6x + 2y = -8, you need to solve the system of equations simultaneously.

First, solve one of the equations for one of the variables:

From the first equation, rearrange it to solve for y:
y = 3x - 4

Then, substitute this expression for y into the second equation:
6x + 2(3x - 4) = -8

Now, solve for x:
6x + 6x - 8 = -8
12x - 8 = -8
12x = 0
x = 0

Now that you have found the value of x, substitute it back into one of the original equations to find y. Let's use the first equation:
3(0) - y = 4
0 - y = 4
y = -4

So, the point of intersection for the two equations is (0, -4).