How do solve the following linear system?: # 9 x-6y =3 , x + y= 12 #?

Question

How do solve the following linear system?: #   9 x-6y =3 ,   x + y= 12  #?

Abigail Allen · Accepted Answer

#x=5# and #y=7#

#9x-6y=3# #x+y=12#

Using the second equation, determine a value for #y#.

#x+y=12#

Subtract #x# from both sides.

#y=12-x#

Reduce the first equation by dividing all terms by #3#.

#9x-6y=3#

#3x-2y=1#

Substitute #y# with #color(red)((12-x))#.

#3x-2color(red)((12-x))=1#

Open the brackets and simplify. The product of two negatives is a positive.

#3x-24+2x=1#

#5x-24=1#

Add #24# to both sides.

#5x=25#

Divide both sides by #5#.

#x=5#

In the second equation, substitute #x# with #color(blue)5#.

#x+y=12#

#color(blue)5+y=12#

Subtract #5# from each side.

#y=7#

Savannah Anderson · Answer

undefined To solve the linear system:

1. Start by solving one of the equations for one variable in terms of the other.
2. Substitute the expression found in step 1 into the other equation.
3. Solve the resulting equation for the variable.
4. Once you have the value of one variable, substitute it back into one of the original equations and solve for the other variable.
5. Verify the solution by substituting the values of both variables into both original equations.

Given the system:

$$9x - 6y = 3$$
$$x + y = 12$$

Let's solve equation 2 for $x$:

$$x = 12 - y$$

Now, substitute this expression for $x$ into equation 1:

$$9(12 - y) - 6y = 3$$

$$108 - 9y - 6y = 3$$

$$108 - 15y = 3$$

$$15y = 105$$

$$y = 7$$

Now, substitute $y = 7$ into equation 2 to solve for $x$:

$$x = 12 - 7$$

$$x = 5$$

So, the solution to the system is $x = 5$ and $y = 7$.