An equation system is a set of two or more equations with two or more variables that need to be solved at the same time. To solve an equation system, we need to find the values of the variables that satisfy all the equations in the system.
There are several methods to solve equation systems, but one of the most common is the elimination method. Here's how it works:
Look at the equations and choose a variable that you want to eliminate. This is usually the variable that has the same coefficient in both equations.
Multiply one or both of the equations by a number that will make the coefficients of that variable opposite in sign. For example, if you want to eliminate x and one equation has a coefficient of 2x and the other has -3x, you can multiply the second equation by 2 to get -6x.
Add or subtract the equations to eliminate the chosen variable. For example, if you multiplied the second equation by 2, you would then add the two equations together to eliminate x.
Solve the resulting equation for the remaining variable.
Substitute this value back into one of the original equations and solve for the other variable.
Check your solution by plugging in the values you found into both equations to make sure they are true.
Here's an example of how to solve an equation system using the elimination method:
Equation 1: 2x + 3y = 12 Equation 2: 4x - 5y = -7
Let's choose to eliminate the x variable. We can do this by multiplying the first equation by -2 to get -4x - 6y = -24.
Add the second equation to the new equation we just made: -4x - 6y = -24 + 4x - 5y = -7.
Simplify: -11y = -31
Solve for y: y = 31/11
Substitute this value back into the first equation and solve for x: 2x + 3(31/11) = 12. Simplifying gives us 2x = (12 - 93/11), so x = 9/11.
Check our solution: plugging in x=9/11 and y=31/11 into both equations gives us true statements.
That's how to solve an equation system using the elimination method!