How do solve the following linear system?: # -x-7y=14 , 3x-2y=2 #?

Question

How do solve the following linear system?: #  -x-7y=14 ,  3x-2y=2    #?

Ethan Adkins · Accepted Answer

Isolate one of the variables in the one of the equations and use that to solve the other variable. Once one of the variable is defined, use it to define the other variable.

Solving linear equations implies finding the intersections of the two equations. There are many ways to solve a linear system given two equations. In my opinion, the easiest is to isolate one of the variables in one equation, and sub it into the other. We'll isolate the first equation since it's almost done. #-x-7y=14# #-x=14+7y# #x=-14-7y# Now we sub in #x=-14-7y# into the other equation. #3x-2y=2# #3(-14-7y)-2y=2# Now we expand the bracket. #(-42-21y)-2y=2# #-42-21y-2y=2# And we add like terms. #-21y-2y=2+42# #-23y=44# Isolate #y#. #y=-44/23# Now we use this value to solve for #x#. We can use any equation, as long as it's the original two. #-x-7y=14# #-x-7(-44/23)=14# #-x+308/23=14# #-x=14-308/23# #x=308/23-14# #x=-14/23# Thus, the linear system intersects at #(-14/23,-44/23)#. We can double check our work by graphing both equations and finding their intersection. graph{(-x-7y-14)(3x-2y-2)=0} Hope this helps :)

Julia Adams · Answer

undefined To solve the linear system:

1. Rearrange the equations so that they are in standard form (Ax + By = C).

Given:
Equation 1: -x - 7y = 14
Equation 2: 3x - 2y = 2

2. Choose a method to solve the system, such as substitution or elimination.

Let's use the elimination method:

3. Multiply Equation 1 by 3 and Equation 2 by -1 to eliminate x when adding them together:

Equation 1 becomes: -3x - 21y = 42
Equation 2 becomes: -3x + 2y = -2

4. Add the equations together:

(-3x - 21y) + (-3x + 2y) = 42 - 2
-21y + 2y = 40
-19y = 40

5. Solve for y:

y = 40 / -19
y ≈ -2.1053

6. Substitute the value of y into one of the original equations to solve for x. Let's use Equation 1:

-x - 7(-2.1053) = 14
-x + 14.7371 = 14
-x ≈ 14 - 14.7371
-x ≈ -0.7371
x ≈ 0.7371

So, the solution to the linear system is approximately x ≈ 0.7371 and y ≈ -2.1053.