Mathematics » Introducing Graphs » Use the Rectangular Coordinate System

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.

Finding Solutions to Linear Equations in Two Variables
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.Finding Solutions to Linear Equations in Two Variables

 

Is the result a true equation?

 

Yes!

Finding Solutions to Linear Equations in Two Variables

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.Finding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two Variables
Simplify.

 

Solve.

Finding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two Variables
 Finding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two Variables
Write the ordered pair.\(\left(2,0\right)\)\(\left(1,\frac{3}{2}\right)\)\(\left(4,-3\right)\)

Check your answers.

 
\(\left(2,0\right)\)\(\left(1,\frac{3}{2}\right)\)\(\left(4,-3\right)\)
Finding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two Variables

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

 Finding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two Variables
Choose a value for \(x\) or \(y.\)Finding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two Variables
Substitute it into the equation.Finding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two Variables
Solve.Finding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two VariablesFinding Solutions to Linear Equations in Two Variables
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.

[Attributions and Licenses]


This is a lesson from the tutorial, Introducing Graphs and you are encouraged to log in or register, so that you can track your progress.

Log In

Ask Question, Post Comment, Tip or Contribution

Do NOT follow this link or you will be banned from the site!