Solving Systems of Linear Equations: A Visual Guide

To solve a system of linear equations graphically, we start with our coordinate plane.We have two equations: y equals two x plus one, and y equals negative x plus four.Let's plot our first equation, y equals two x plus one, shown in blue. We'll plot several points to help us draw the line accurately.Now for our second equation, y equals negative x plus four, shown in red. Notice how this line slopes downward.The point where these lines intersect is our solution. At this point, both equations are satisfied simultaneously.Let's verify this solution by plugging the point (1, 3) into both equations.Therefore, the point (1, 3) is the solution to our system of equations.Now that we've found our solution graphically, let's explore other methods to verify this result.The substitution method solves systems of equations by replacing one variable with an expression.We start with equation one, which is already solved for y. This makes it perfect for substitution.We take the expression for y from equation one, and substitute it into equation two.Next, we move all terms with x to the left side of the equation.Now we can combine like terms. Two x plus x gives us three x.Subtract one from both sides.Finally, divide both sides by three to solve for x.Now that we know x equals one, we can substitute this back into either of our original equations. Let's use equation one.Two times one is two.Add one to get our y-value of three.Let's verify our solution in both original equations.In equation one, when x is one, y equals three.And in equation two, negative one plus four also equals three.Now we'll explore the elimination method, a powerful technique for solving systems of equations.Notice that these equations have opposite coefficients for y: positive 3y in equation 1 and negative 3y in equation 2. This makes them perfect for elimination.When we add these equations together, the y terms will cancel out completely.The first equation already has the right coefficient for y, so we don't need to multiply it by anything.Now, when we add the equations, the y terms eliminate each other, leaving us with just x terms.This gives us six x equals eighteen, which means x equals three.Now that we know x equals three, we can substitute this back into either original equation. Let's use the first equation.Let's verify our solution by checking both original equations with x equals three and y equals two.The elimination method is particularly powerful when coefficients can be easily aligned, especially when they're already opposites like in this example.Thanks for learning about solving systems of equations with Spark.E!