# Expressions with One or More Variables

## Key Questions

• There are many methods that you can use for solving equations with two variables. Here are two simple methods that I find easy for solving linear equations with two variables.

Remember, you always need at least two equations to solve equations with two variables

1. Substitution
Here, you basically write one equation in terms of one of its variables, and then substitute that value in the second equation.

Example:
Solving two equations:
$3 x + y = 5$
$2 x + 3 y = 4$

Now, I'll write the first equation in terms of $y$. (I can also do it in terms of $x$.)
$y = 5 - 3 x$

Now, I can substitute this value of $y$ in the second equation as follows:
$2 x + 3 \left(5 - 3 x\right) = 4$

$2 x + 15 - 9 x = 4$

$- 7 x = - 11$

$x = \frac{- 11}{-} 7$

$x = \frac{11}{7}$

Thus,

$y = 5 - 3 \left(\frac{11}{7}\right) = 5 - \frac{33}{7} = \frac{2}{7}$

2. Elimination
Here, you:
a. First, multiply one or both equations so that either variable have the same coefficients.
b. Then, add or subtract one equation to/from another so that one of the variable term is completely eliminated.
c. Substitute the value you find out in any other equation to find the value of the other variable.

Example:
Solving two equations:
$3 x + y = 5$
$2 x + 3 y = 4$

Looking at the two equations, I can make out that by multiplying the first equation by $3$, both equations will have the term $3 y .$

The first equation becomes:
$\left(3 x + y = 5\right) \cdot 3$
$9 x + 3 y = 15$

When I subtract the second equation from the above equation, the terms with $y$ will be eliminated.
$9 x + 3 y = 15$
$2 x + 3 y = 4$
$- - - - -$
$7 x + 0 y = 11$

That is,

$7 x = 11$
$x = \frac{11}{7}$

Substituting the value of $x$ in the first equation,

$3 \left(\frac{11}{7}\right) + y = 5$
$y = 5 - \frac{33}{7}$
$y = \frac{2}{7}$

SIMILARLY, YOU CAN EVALUATE EXPRESSIONS WITH MORE VARIABLES WITH MORE NUMBER OF EQUATIONS.

something like 2x+3y

#### Explanation:

A term is a number or a letter or a combination.

If you put them together like 2x+3y this is an expression

If you have an equal sign it becomes an equation like 2x+3y=7

If the LHS equals the RHS it is an identity like 2(x+3)=2x+6

$x + y$