As we move beyond single equations, we encounter situations where two different variables—often x and y—interact within two separate equations. These are known as Simultaneous Equations. Solving them means finding the specific pair of values that makes both equations true at the same time.
In this chapter, Wentworth introduces the three primary methods for solving these systems:
If we have a single equation with two unknown quantities, such as x + y = 10, there are an infinite number of solutions (e.g., 5+5, 6+4, 12-2). However, if we introduce a second independent equation, such as x - y = 2, there is only one set of values that satisfies both: x = 6 and y = 4.
To solve these, we use Elimination, which is the process of combining the two equations in a way that causes one of the variables to disappear, leaving us with a simple equation in one variable.
Multiply one or both equations by numbers that make the coefficients of one variable (say, x) equal. If the signs are different, add the equations; if the signs are the same, subtract them.
From one of the equations, find the value of one unknown in terms of the other. Then, substitute this value into the other equation.
Find the value of the same unknown in each equation, and then form a new equation by setting these two values equal to each other.