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# Finding Solutions to Linear Equations in Two Variables

## Finding 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 $$1,100,1,000,$$ 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.

## Example

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 $$\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}$$ or any value for $$\phantom{\rule{0.2em}{0ex}}y.$$ Let’s pick $$x=0.$$ What is the value of $$\phantom{\rule{0.2em}{0ex}}y\phantom{\rule{0.2em}{0ex}}$$ if $$\phantom{\rule{0.2em}{0ex}}x=0$$? Step 2: Substitute that value into the equation. Solve for the other variable. Substitute $$0$$ for $$\phantom{\rule{0.2em}{0ex}}x.$$ Simplify.  Divide both sides by 2. Step 3: Write the solution as an ordered pair. So, when $$x=0,y=3.$$ This solution is represented by the ordered pair $$\left(0,3\right).$$ Step 4: Check. Is the result a true equation? Yes!

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 $$3x+2y=6.$$

## Example

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 $$-3$$ for $$y.$$

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

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

So $$\left(2,0\right),\left(1,\frac{3}{2}\right)$$ and $$\left(4,-3\right)$$ 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.

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

Let’s find some solutions to another equation now.

## Example

Find three solutions to the equation $$x-4y=8.$$

### Solution

 Choose a value for $$x$$ or $$y.$$ Substitute it into the equation. Solve. Write the ordered pair. $$\left(0,-2\right)$$ $$\left(8,0\right)$$ $$\left(20,3\right)$$

So $$\left(0,-2\right),\left(8,0\right),$$ and $$\left(20,3\right)$$ are three solutions to the equation $$x-4y=8.$$

$$x-4y=8$$
$$x$$$$y$$$$\left(x,y\right)$$
$$0$$$$-2$$$$\left(0,-2\right)$$
$$8$$$$0$$$$\left(8,0\right)$$
$$20$$$$3$$$$\left(20,3\right)$$

Remember, there are an infinite number of solutions to each linear equation. Any point you find is a solution if it makes the equation true.