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How can two linear equations be solved together by substitution?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — algebra This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

To solve a system of two linear equations using substitution, follow these steps to find the values of the variables that satisfy both equations simultaneously.12

Isolate a variable: Choose one of the two equations and solve it for one of the variables (e.g., solve for $y$ in terms of $x$ ).23
Substitute: Take the expression obtained in the first step and substitute it into the other equation in place of that variable. This results in a single equation containing only one unknown variable.23
Solve: Solve this new equation to find the numerical value of the remaining variable.2
Back-substitute: Substitute the value you just found back into one of the original equations to determine the value of the first variable.23

This algebraic method is particularly useful for simple systems of two equations in two unknowns. While graphing can visualize the intersection of two lines, substitution provides a precise numerical solution that avoids the potential inaccuracies of visual estimation. For larger systems, these principles are extended through Gauss elimination or matrix operations.123