Equations with two unknowns represent a relationship between two variables. Equations in two variables often express a relationship between the variables $x$ and $y$, which correspond to Cartesian coordinates. 

Equations in two variables have not one solution but a series of solutions that will satisfy the equation for both variables. Each solution is an ordered pair and can be written in the form $(x, y)$.

Solving Equations in Two Variables

For a given equation in two variables, choosing a value for one variable dictates what the value of the other variable will be. In other words, if a value for one variable is provided, then a solution can be found that satisfies the equation. This is accomplished by substituting the given value in for that variable, and solving for the value of the other.

Example 1

Consider the following equation: 

$y = 2x$

This is an equation in two variables that has an infinite number of solutions. For any $x$-value, the corresponding $y$-value will be twice its value. 

For example, $(1, 2)$ is a solution to the equation. This can be verified by plugging in the $x$- and $y$-values:

$(2) = 2(1)$

Another solution is $(30, 60)$, because $(60) = 2(30)$. There are thus an infinite number of ordered pairs that satisfy the equation.$$

Example 2

Now consider the following equation:

$y = 2x + 4$

Is the point $(3, 10)$ a solution to this equation?

Note that the ordered pair $(3, 10)$ tells us that $x = 3$ and $y = 10$. To evaluate whether this is a solution to the equation, substitute these values in for the variables as follows:

$(10) = 2(3) + 4$

$10 = 6 + 4$

This is a true statement, so $(3, 10)$ is indeed a solution to this equation.

Example 3

Solve the equation $y = 4x - 7$ for the value $x=3$.

The solution to the given equation would take the form $(x, y)$, and we are given the $x$-value. The $x$-value can be substituted into the equation to find the value of $y$ at this point:

$y = 4(3) - 7$

$y = 12 - 7$

$y = 5$

For the given equation, $y = 5$ when $x = 3$. Therefore, the solution is $(3, 5)$.

Example 4

Solve $x + 2y = 8$ for $x = 4$.

As in the above example, the $x$-value is provided, and we need to find the corresponding $y$-value. We can first rewrite the equation in terms of $y$:

$x + 2y -x = 8 -x$

$2y = 8 - x$

$\dfrac{2y}{2} = \dfrac{8-x}{2}$

$y = \dfrac{8}{2} - \dfrac{x}{2}$

$y = 4 - \dfrac{1}{2} x$

Now substitute $x = 4$ into the equation, and solve for $y$: 

$y = 4 - \dfrac{1}{2}(4)$

$y = 4 - 2$

$y = 2$

The solution is $(4, 2)$.