Understanding System Substitution in Linear Equations

Welcome to understanding system substitution! Today we'll explore a powerful method for solving systems of linear equations.A system of linear equations consists of two or more equations that we need to solve simultaneously. Let's look at an example.These equations represent two lines in a coordinate plane. Each line shows all possible values of x and y that satisfy that equation.System substitution is a method where we take one equation that's solved for a variable, and use it to replace that variable in the other equation.This method works particularly well when one equation is already solved for a variable, like our first equation y equals two x plus one.Notice how the first equation has y isolated on one side. This makes it perfect for substitution.When choosing which equation to use for substitution, look for one that's already solved for a variable. This form makes the substitution process much easier.Now that we understand what system substitution is and when to use it, let's see how to perform the substitution.We start with our system of equations.The first equation is already solved for y, making it perfect for substitution.We'll substitute this expression for y into the second equation.First, we identify the expression for y.Then, we substitute this expression wherever y appears in the second equation.When we make this substitution, we replace y with two x plus one, being careful to use parentheses.Next, we can remove the parentheses and combine like terms. Three x plus two x gives us five x.Finally, we simplify the expression to get five x plus one equals seven.Let's break down how we combined like terms. Three x plus two x combines to give us five x, while the constant term one remains unchanged.Remember these important points when performing substitution: always use parentheses, combine like terms carefully, and verify each step.Now that we have our equation with one variable, we're ready to solve for x.Now that we have our simplified equation, let's solve for x.First, subtract 1 from both sides to isolate the term with x.Then divide both sides by 5 to solve for x.Now that we know x equals 1.2, we can substitute this value back into either of our original equations. Let's use y equals 2x plus 1.Let's verify our solution by checking both original equations.First, let's check y equals 2x plus 1.Then, let's verify 3x plus y equals 7.Let's review some key points about verifying our solution.Our final solution to the system is x equals 1.2 and y equals 3.4.Remember to always verify your solutions in both original equations to ensure accuracy.Thanks for learning about solving systems of equations!