![]() ![]() We eliminated both variables and arrived at a true statement. Substitute y = 2x - 4 into the first equation: Now, we will see how to solve the following system of linear equations using the substitution method: Substitute y = -2 into the first equation: Can be used to prove both upper bounds O() and lower bounds (). Substitute x = -1.5y + 2.5 into the second equation: 1 Solving Recurrences with the Substitution Method Idea: Make a guess for the form of the solution and prove by induction. Substitute y = 1.5 into the second equation: You can then solve this equation as it will now have only one variable. Then substitute that expression in place of that variable in the second equation. To use the substitution method, use one equation to find an expression for one of the variables in terms of the other variable. Substitute 4y - 2 for x into the first equation: The substitution method is one way of solving systems of equations. Use the substitution method to solve the system of equations: In this section, we show you step by step how to solve several systems using the substitution method so that you can see how to do the substitution method in practice. If the statement is true, then the system is dependent, i.e., it has infinitely many solutions. ![]() If the statement is false, then the system is inconsistent, which means it has no solution. All you need to do is draw conclusions about the system depending on whether the statement you got is true (like 0 = 0 or 17 = 17) or false (e.g., 0 = 1 or 15 = 17): ![]() ⚠️ It may sometimes happen that you try to solve a system, and suddenly both variables vanish □ Keep calm! The variables are no more, but you arrived at some statements about numbers. This way, we obtain an equation with one variable, which we can easily solve. The main idea behind solving systems with the substitution method is to choose one of the equations, solve it for one of the variables, and plug the result into the other equation. If you want, you may test your solution: substitute the values you obtained into the system and see if everything is OK. The substitution method is a method of solving systems of linear equations. That's it! You solved the system of equations by the substitution method. Substitute the value you got in Step 5 into one of the original equations. You have obtained a one-variable equation – solve it! This is the essence of solving systems by the substitution method! Plug the result into the other equation (the one you didn't choose in Step 1). In the chosen equation, choose one of the variables. Now, let's discuss in detail how to do the substitution method: The substitution method is a method of solving systems of linear equations. We have already explained what the substitution method is about and what the main idea behind it is. ![]()
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