What is the name for a system of equations with the same variables?

A system of equations with shared variables is commonly called a set of simultaneous equations. The equations are considered together because the goal is to find values that satisfy all of them at once.

For example, x + y = 10 and x − y = 2 form simultaneous equations in x and y. Solving them together gives x = 6 and y = 4. Methods such as substitution, elimination, and graphing can reveal the common solution.

“Linear equations” is a related but narrower term. It describes equations whose variables have a first degree, such as 2x + 3y = 7. A system of simultaneous equations may be linear, but it can also include nonlinear equations, such as x² + y = 5. “Differential equations” involve derivatives, while quadratic equations contain a second-degree variable; neither term simply means equations sharing variables.

The phrase “simultaneous equations” is especially common in British and Commonwealth mathematics, while “system of equations” is widely used in American textbooks. Both refer to equations solved as a connected group.