11.1 Use the rectangular coordinate system  (Page 5/13)

We’ll start by looking at the solutions to the equation $y=5x-1$ we found in [link] . We can summarize this information in a table of solutions.

To find a third solution, we’ll let $x=2$ and solve for $y.$

	$y=5x-1$
Multiply.	$y=10-1$
Simplify.	$y=9$

The ordered pair is a solution to $y=5x-1$ . We will add it to the table.

We can find more solutions to the equation by substituting any value of $x$ or any value of $y$ and solving the resulting equation to get another ordered pair that is a solution. There are an infinite number of solutions for this equation.

Complete the table to find three solutions to the equation $y=4x-2\text{:}$

$y=4x-2$
$x$	$y$	$(x,y)$
$0$
$\mathrm{-1}$
$2$

Solution

Substitute $x=0,x=\mathrm{-1},$ and $x=2$ into $y=4x-2.$

$y=4x-2$	$y=4x-2$	$y=4x-2$
$y=0-2$	$y=\mathrm{-4}-2$	$y=8-2$
$y=\mathrm{-2}$	$y=\mathrm{-6}$	$y=6$
$(0,\mathrm{-2})$	$(\mathrm{-1},\mathrm{-6})$	$(2,6)$

The results are summarized in the table.

$y=4x-2$
$x$	$y$	$(x,y)$
$0$	$\mathrm{-2}$	$(0,\mathrm{-2})$
$\mathrm{-1}$	$\mathrm{-6}$	$(\mathrm{-1},\mathrm{-6})$
$2$	$6$	$(2,6)$

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Complete the table to find three solutions to the equation $5x-4y=20\text{:}$

$5x-4y=20$
$x$	$y$	$(x,y)$
$0$
	$0$
	$5$

Solution

The results are summarized in the table.

$5x-4y=20$
$x$	$y$	$(x,y)$
$0$	$\mathrm{-5}$	$(0,\mathrm{-5})$
$4$	$0$	$(4,0)$
$8$	$5$	$(8,5)$

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Find solutions to linear equations in two variables

To find a solution to a linear equation, we can choose any number we want to substitute into the equation for either $x$ or $y.$ We could choose $1,100,\mathrm{1,000},$ or any other value we want. But it’s a good idea to choose a number that’s easy to work with. We’ll usually choose $0$ as one of our values.

Find a solution to the equation $3x+2y=6.$

Solution

Step 1: Choose any value for one of the variables in the equation.		We can substitute any value we want for $x$ or any value for $y.$ Let's pick $x=0.$ What is the value of $y$ if $x=0$ ?
Step 2: Substitute that value into the equation. Solve for the other variable.	Substitute $0$ for $x.$ Simplify. Divide both sides by 2.
Step 3: Write the solution as an ordered pair.	So, when $x=,y=3.$	This solution is represented by the ordered pair $(0,3).$
Step 4: Check.	Is the result a true equation? Yes!

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We said that linear equations in two variables have infinitely many solutions, and we’ve just found one of them. Let’s find some other solutions to the equation $3x+2y=6.$

Find three more solutions to the equation $3x+2y=6.$

Solution

To find solutions to $3x+2y=6,$ choose a value for $x$ or $y.$ Remember, we can choose any value we want for $x$ or $y.$ Here we chose $1$ for $x,$ and $0$ and $\mathrm{-3}$ for $y.$

Substitute it into the equation.
Simplify. Solve.
Write the ordered pair.	$(2,0)$	$(1,\frac{3}{2})$	$(4,\mathrm{-3})$

Check your answers.

$(2,0)$	$(1,\frac{3}{2})$	$(4,\mathrm{-3})$

So $\left(2,0\right),\left(1,\frac{3}{2}\right)$ and $\left(4,\mathrm{-3}\right)$ are all solutions to the equation $3x+2y=6.$ In the previous example, we found that $\left(0,3\right)$ is a solution, too. We can list these solutions in a table.

$3x+2y=6$
$x$	$y$	$(x,y)$
$0$	$3$	$(0,3)$
$2$	$0$	$(2,0)$
$1$	$\frac{3}{2}$	$(1,\frac{3}{2})$
$4$	$\mathrm{-3}$	$(4,\mathrm{-3})$

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