How do you solve 3x + y = 7 and 3x - y = 5?

Jun 7, 2016

$x = 2 \text{ and } y = 1$

Explanation:

There are 2 obvious methods which can be used in this example, based on the x and y values in the equations.

ELIMINATION METHOD
The two y terms are additive inverses. This means they have the same value, but opposite signs.
When they are ADDED together, they will make 0.

$\text{ " 3x + y = 7 " A}$
$\text{ " 3x - y = 5 " B}$

$\text{A + B: " 6x " "= 12" (the y's have been eliminated)}$
$\text{ " x" } = 2$

Substitute into A to find the value of y:
$\text{ } 3 \left(2\right) + y = 7$
$\text{ } 6 + y = 7$
$\text{ } y = 1$

EQUATING METHOD

The two x-terms are exactly the same. Make the x term the subject.

$3 x + y = 7 \text{ and } 3 x - y = 5$
$3 x \text{ " = 7 - y " " 3x " } = 5 + y$

But $3 x \text{ is the same as } 3 x$!!

Therefore $\text{ } 7 - y = 5 + y$
solve for y: $\text{ } 7 - 5 = 2 y$
$\text{ " 2 = 2y " } \Rightarrow y = 1$

If $y = 1$, substitute into either equation to find $x$.

$3 x = 7 - y \Rightarrow 3 x = 7 - 1 = 6$
$3 x = 6$
$x = 2$