How do you solve the following system using substitution?: #3(x+5) - 2(y - 1) = 6, x + 4(y + 3) = 13#

Question

Julia Adams · Accepted Answer

#x = -3#, #y = 1#

To solve the system using substitution, first we need to transform one of the equations such that one of the variables is expressed as an equality in terms of the other variable(s).

#[1] 3(x +5) - 2(y-1) = 6# #[2] x + 4(y+3) = 13#

Let us isolate #x# in equation #[2]#

#[2] x + 4(y+3) = 13# #[2] => x = 13 - 4(y + 3)# #[2] => x = 13 - 4y - 12# #[2] => x = -4y + 1#

We then substitute #x# in equation #[1]# with its equivalent we obtained in equation #[2]#.

#[1] 3(x + 5) - 2(y - 1) = 6#

#[1] => 3((-4y + 1) + 5) - 2(y-1) = 6# #[1] => 3(-4y + 6) -2(y-1) = 6# #[1] => -12y + 18 -2y + 2 = 6# #[1] => -14y = 6 - 18 - 2# #[1] => -14y = -14# #[1] => y = 1#

To get #x#, replace #y# with its value in any of the equations above

#x = -4y + 1# #x = -4(1) + 1#

#x = -4 + 1#

#x = -3#

Hazel Anglin · Answer

undefined To solve the system using substitution:
1. Solve one of the equations for one variable.
2. Substitute this expression into the other equation.
3. Solve for the remaining variable.
4. Substitute the found value back into one of the original equations to find the value of the other variable.

Given equations:
1. 3(x+5) - 2(y - 1) = 6
2. x + 4(y + 3) = 13

Step 1: Solve equation 1 for x:
3(x+5) - 2(y - 1) = 6
=> 3x + 15 - 2y + 2 = 6
=> 3x - 2y + 17 = 6
=> 3x - 2y = -11
=> x = (2y - 11) / 3

Step 2: Substitute x from equation 1 into equation 2:
x + 4(y + 3) = 13
=> (2y - 11) / 3 + 4(y + 3) = 13

Step 3: Solve for y:
=> (2y - 11) / 3 + 4(y + 3) = 13
=> (2y - 11) + 12(y + 3) = 39
=> 2y - 11 + 12y + 36 = 39
=> 14y + 25 = 39
=> 14y = 14
=> y = 1

Step 4: Substitute y = 1 into equation 1 to find x:
3x - 2(1) = -11
=> 3x - 2 = -11
=> 3x = -9
=> x = -3

So, the solution to the system is x = -3 and y = 1.