Solving Systems by Substitution: A Simple Guide

Welcome to understanding the substitution method for solving systems of equations. Today we'll focus on the crucial first step: isolating variables.Let's start with two equations. Notice how the second equation already has y isolated on one side.When solving systems of equations using substitution, our first goal is to get one variable by itself on one side of the equation.We need to carefully choose which variable to isolate based on which will make our substitution easier.Then we move all terms with our chosen variable to one side of the equation.Finally, we solve for that variable to get it completely isolated.Let's see this process in action with our first equation.First, we rearrange the terms, moving y next to the equals sign.Then we subtract 2x from both sides to isolate y.Finally, we can rewrite this in standard form with the variable terms first.Now let's look at some guidelines for choosing which variable to isolate.Choose a variable that has simpler coefficients to make your calculations easier.Make sure the variable you choose appears in both equations, as we'll need to substitute it later.When possible, avoid creating fractions, as they can make calculations more complicated.Now that we understand how to isolate variables, we're ready to move on to the actual substitution process.Now that we have our isolated variable from equation one, let's perform the substitution into equation two.We'll replace y in the second equation with the expression two x plus one from the first equation.When we make this substitution, it's crucial to use parentheses to maintain the correct order of operations.Let's look at some common errors to avoid when performing substitution.After making the substitution, we need to distribute the negative sign correctly.Let's try another example to reinforce proper substitution technique.Here, we substitute x equals y minus three into the second equation, being careful with our parentheses.Now that we've performed our substitution correctly, we're ready to solve the resulting equation.Now that we have our equation with one variable, let's solve it step by step.First, distribute the negative sign and combine like terms.Adding one to both sides gives us x equals 6.Now we can substitute x equals 6 back into the equation y equals 2x plus 1 to find y.Let's verify our solution by checking both original equations.For the first equation, three times six minus thirteen equals five.For the second equation, thirteen equals two times six plus one.We can visualize our solution point at coordinates six comma thirteen on the coordinate plane.