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:

$y=\dfrac{3}{2}$

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

$x + 3y = 7\\\\x + 3(\dfrac{3}{2}) = 7\\\\x + \dfrac{9}{2} = 7\\\\x = 7- \dfrac{9}{2}\\\\x = 7- 4.5\\\\x = 2.5\\\\x=\dfrac{5}{2}$

Answer

$x=\dfrac{5}{2}$

$y=\dfrac{3}{2}$