4.1 Use the rectangular coordinate system  (Page 3/12)

Name the ordered pair of each point shown in the rectangular coordinate system.

A: $(5,1)$  B: $(\mathrm{-2},4)$  C: $(\mathrm{-5},\mathrm{-1})$  D: $(3,\mathrm{-2})$  E: $(0,\mathrm{-5})$  F: $(4,0)$

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Name the ordered pair of each point shown in the rectangular coordinate system.

A: $(4,2)$  B: $(\mathrm{-2},3)$  C: $(\mathrm{-4},\mathrm{-4})$  D: $(3,\mathrm{-5})$  E: $(\mathrm{-3},0)$  F: $(0,2)$

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Verify 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.

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 .

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

The equation $y=\mathrm{-3}x+5$ 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.

$\begin{array}{cccccc}& & & \hfill y& =\hfill & \mathrm{-3}x+5\hfill \\ \text{Add to both sides.}\hfill & & & \hfill y+3x& =\hfill & \mathrm{-3}x+5+3x\hfill \\ \text{Simplify.}\hfill & & & \hfill y+3x& =\hfill & 5\hfill \\ \begin{array}{c}\text{Use the Commutative Property to put it in}\hfill \\ Ax+By=C\text{form}.\hfill \end{array}\hfill & & & \hfill 3x+y& =\hfill & 5\hfill \end{array}$

By rewriting $y=\mathrm{-3}x+5$ 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 .

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 $(x,y)$ . 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.