Algebra
Textbooks
Boundless Algebra
Introduction to Equations, Inequalities, and Graphing
Graphing and Equations of Two Variables
Algebra Textbooks Boundless Algebra Introduction to Equations, Inequalities, and Graphing Graphing and Equations of Two Variables
Algebra Textbooks Boundless Algebra Introduction to Equations, Inequalities, and Graphing
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 4
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Equations in Two Variables

Equations with two unknowns represent a relationship between two variables and have a series of solutions.

Learning Objective

  • Explain what an equation in two variables represents


Key Points

    • An equation in two variables has a series of solutions that will satisfy the equation for both variables. 
    • Each solution to an equation in two variables is an ordered pair and can be written in the form $(x, y)$. 

Terms

  • Cartesian coordinates

    The coordinates of a point measured from an origin along a horizontal axis from left to right (the $x$-axis) and along a vertical axis from bottom to top (the $y$-axis). 

  • ordered pair

    A set containing exactly two elements in a fixed order, used to represent a point in a Cartesian coordinate system. Notation: $(x, y)$.


Full Text

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 $(4, 2)$.

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