1.5 Equivalent Equations
1.5.1) Equivalent Equations with Two Variables
Equivalent equations with two variables are equations that have the same set of solutions. For example, the equations \(2x + 3y = 6\) and \(4x + 6y = 12\) are equivalent because they represent the same line in the coordinate plane, and thus have the same set of solutions.
1.5.2) Creating Equivalent Equations with Two Variables
We can create equivalent equations by multiplying or dividing both sides of an equation by the same non-zero quantity.
Example: If we multiply both sides of the equation \(2x + 3y = 6\) by 2, we get \(4x + 6y = 12\), which is an equivalent equation.
1.5.3) Checking for Equivalence with Two Variables
Method 1: Graph Both Equations
One way to check if two equations are equivalent is to graph both equations and see if they represent the same line.
Method 2: Substitute Values
Another way is to substitute the same values for the variables in both equations and see if the results are the same.
Practice Problems
- Are the equations \(2x + 3y = 6\) and \(4x + 6y = 12\) equivalent?
- Create an equivalent equation for \(3x + 2y = 14\).
- Check if the equations \(4x + 2y = 10\) and \(2x + y = 5\) are equivalent.