In solving systems of equations, the substitution method provides a systematic approach to find the values of variables without the need for graphing. This method is particularly useful when equations are complex or when graphing is impractical. The solution to a system of equations consists of a pair of numbers (x, y) that satisfy both equations simultaneously.
The first step in the substitution method is to identify the easier equation to isolate one of the variables, typically denoted as equation A. For instance, if you have the equation y = 7x - 14, it is already solved for y, making it a suitable choice for equation A. The other equation will be designated as equation B.
Next, you will substitute the expression from equation A into equation B. This means that wherever the variable y appears in equation B, you will replace it with the expression from equation A. For example, if equation B is 2x - y = 4, substituting gives you 2x - (7x - 14) = 4. This substitution simplifies the equation to a single variable, allowing you to solve for x.
After performing the substitution, you will simplify the equation. Continuing with our example, you would distribute the negative sign and combine like terms, resulting in -5x + 14 = 4. Solving this equation leads to x = 2.
With the value of x determined, the next step is to find y by substituting x back into either equation A or B. Using equation A, you would substitute x = 2 into y = 7(2) - 14, which simplifies to y = 0. Thus, the solution to the system of equations is (x, y) = (2, 0).
Finally, it is essential to verify your solution by substituting both values back into the original equations. For equation A, substituting gives 0 = 7(2) - 14, which holds true. For equation B, substituting yields 2(2) - 0 = 4, confirming the solution is correct. This verification step ensures that the values found satisfy both equations, solidifying the accuracy of the substitution method.
