Finding Solutions to Linear Equations in Two Variables
Contents
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. | \(\left(2,0\right)\) | \(\left(1,\frac{3}{2}\right)\) | \(\left(4,-3\right)\) |
Check your answers.
| \(\left(2,0\right)\) | \(\left(1,\frac{3}{2}\right)\) | \(\left(4,-3\right)\) |
| | |
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.