Verifying Solutions to An Equation in Two Variables

Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. The process of solving an equation ended with a statement like \(x=4\). (Then, you checked the solution by substituting back into the equation.)

Here’s an example of an equation in one variable, and its one solution.

\(\begin{array}{ccc}\hfill 3x+5& =\hfill & 17\hfill \\ \hfill 3x& =\hfill & 12\hfill \\ \hfill x& =\hfill & 4\hfill \end{array}\)

But equations can have more than one variable. Equations with two variables may be of the form \(Ax+By=C\). Equations of this form are called linear equations in two variables.

Linear Equation

An equation of the form \(Ax+By=C\), where \(A\) and \(B\) are not both zero, is called a linear equationin two variables.

Notice the word line in linear. Here is an example of a linear equation in two variables, \(x\) and \(y\).

In this figure, we see the linear equation Ax plus By equals C. Below this is the equation x plus 4y equals 8. Below this are the values A equals 1, B equals 4, and C equals 8.

The equation \(y=-3x+5\) is also a linear equation. But it does not appear to be in the form \(Ax+By=C\). We can use the Addition Property of Equality and rewrite it in \(Ax+By=C\) form.

\(\begin{array}{cccccc}& & & \hfill \phantom{\rule{5em}{0ex}}y& =\hfill & -3x+5\hfill \\ \text{Add to both sides.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}y+3x& =\hfill & -3x+5+3x\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}y+3x& =\hfill & 5\hfill \\ \begin{array}{c}\text{Use the Commutative Property to put it in}\hfill \\ Ax+By=C\phantom{\rule{0.5em}{0ex}}\text{form}.\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}3x+y& =\hfill & 5\hfill \end{array}\)

By rewriting \(y=-3x+5\) as \(3x+y=5\), we can easily see that it is a linear equation in two variables because it is of the form \(Ax+By=C\). When an equation is in the form \(Ax+By=C\), we say it is in standard form.

Standard Form of Linear Equation

A linear equation is in standard form when it is written \(Ax+By=C\).

Most people prefer to have \(A\), \(B\), and \(C\) be integers and \(A\ge 0\) when writing a linear equation in standard form, although it is not strictly necessary.

Linear equations have infinitely many solutions. For every number that is substituted for \(x\) there is a corresponding \(y\) value. This pair of values is a solution to the linear equation and is represented by the ordered pair \(\left(x,y\right)\). When we substitute these values of \(x\) and \(y\) into the equation, the result is a true statement, because the value on the left side is equal to the value on the right side.

Solution of a Linear Equation in Two Variables

An ordered pair\(\left(x,y\right)\) is a solution of the linear equation \(Ax+By=C\), if the equation is a true statement when the x- and y-values of the ordered pair are substituted into the equation.

Example

Determine which ordered pairs are solutions to the equation \(x+4y=8\).

a. \(\left(0,2\right)\)
b. \(\left(2,-4\right)\)
c. \(\left(-4,3\right)\)

Solution

Substitute the x- and y-values from each ordered pair into the equation and determine if the result is a true statement.

This figure has three columns. At the top of the first column is the ordered pair (0, 2). Below this are the values x equals 0 and y equals 2. Below this is the equation x plus 4y equals 8. Below this is the same equation with 0 and 2 substituted for x and y: 0 plus 4 times 2 might equal 8. Below this is 0 plus 8 might equal 8. Below this is 8 equals 8 with a check mark next to it. Below this is the sentence “(0, 2) is a solution.” At the top of the second column is the ordered pair (2, negative 4). Below this are the values x equals 2 and y equals negative 4. Below this is the equation x plus 4y equals 8. Below this is the same equation with 2 and negative 4 substituted for x and y: 2 plus 4 times negative 4 might equal 8. Below this is 2 plus negative 16 might equal 8. Below this is negative 14 does not equal 8. Below this is the sentence: “(2, negative 4) is not a solution.” At the top of the third column is the ordered pair (negative 4, 3). Below this are the values x equals negative 4 and y equals 3. Below this is the equation x plus 4y equals 8. Below this is the same equation with negative 4 and 3 substituted for x and y: negative 4 plus 4 times 3 might equal 8. Below this is negative 4 plus 12 might equal 8. Below this is 8 equals 8 with a check mark next to it. Below this is the sentence: “(negative 4, 3) is a solution.”

Example

Which of the following ordered pairs are solutions to the equation \(y=5x-1\)?

a. \(\left(0,-1\right)\)
b. \(\left(1,4\right)\)
c. \(\left(-2,-7\right)\)

Solution

Substitute the x- and y-values from each ordered pair into the equation and determine if it results in a true statement.

This figure has three columns. At the top of the first column is the ordered pair (0, negative 1). Below this are the values x equals 0 and y equals negative 1. Below this is the equation y equals 5x minus 1. Below this is the same equation with 0 and negative 1 substituted for x and y: negative 1 might equal 5 times 0 minus 1. Below this is negative 1 might equal 0 minus 1. Below this is negative 1 equals negative 1 with a check mark next to it. Below this is the sentence: “(0, negative 1) is a solution.” At the top of the second column is the ordered pair (1, 4). Below this are the values x equals 1 and y equals 4. Below this is the equation y equals 5x minus 1. Below this is the same equation with 1 and 4 substituted for x and y: 4 might equal 5 times 1 minus 1. Below this is 4 might equal 5 minus 1. Below this is 4 equals 4 with a check mark next to it. Below this is the sentence: “(1, 4) is a solution.” At the top of the right column is the ordered pair (negative 2, negative 7). Below this are the values x equals negative 2 and y equals negative 7. Below this is the equation y equals 5x minus 1. Below this is the same equation with negative 2 and negative 7 substituted for x and y: negative 7 might equal 5 times negative 2 minus 1. Below this is negative 7 might equal negative 10 minus 1. Below this is negative 7 does not equal negative 11. Below this is the sentence: “(negative 2, negative 7) is not a solution.”

Verifying Solutions to An Equation in Two Variables