Question

Solve the following linear system algebraically. State why you chose the method you used.

x + 3y = 7
2x + 4y = 11

Answer 1

System of Equations

There are two methods of solving systems of equations:

• substitution
• elimination

Substitution is where we substitute one equation into the other by isolating a certain variable, or a group of terms.

Elimination is where we subtract the two equations. Before doing this, we may have to multiply one equation by a certain number to make sure one variable cancels out.

Solving the Question

We're given the following equations:

• 
  `x + 3y = 7`
• 
  `2x + 4y = 11`

Because they are organized in the same manner (i.e. x [operation] y [equals] number), it is easier for us to use elimination.

First, multiply the first equation by 2:

`x + 3y = 7\\2(x + 3y) = 2(7)\\2x + 6y = 14`

Now, subtract the second equation from the one we just created:

`\hspace{10}2x + 6y = 14\\-2x + 4y = 11\\\rule{67}{0.3}\\2y=3`

Solve for y:

`y=(3)/(2)`

To solve for x, we can use substitution in the first equation:

`x + 3y = 7\\\\x + 3((3)/(2)) = 7\\\\x + (9)/(2) = 7\\\\x = 7- (9)/(2)\\\\x = 7- 4.5\\\\x = 2.5\\\\x=(5)/(2)`

Answer

`x=(5)/(2)`

`y=(3)/(2)`