Solving Mixed Systems of Equations with Substitution: A Simple Guide

Mixed systems of equations combine different types of equations to solve complex problems.In our example, we'll work with two equations: a linear equation and a quadratic equation.The first equation is y equals x plus 2. This is a linear equation, meaning it forms a straight line when graphed.The second equation is x squared plus y equals 5. This is a quadratic equation, which creates a curved line when graphed.When we graph both equations together, we can see they intersect at two points. These intersection points represent the solutions to our mixed system.Let's note some important characteristics of mixed systems. They combine different types of equations, require solutions that satisfy both equations, and can have zero, one, or two solutions.We start with our system of equations: y equals x plus 2, and x squared plus y equals 5.Since y equals x plus 2 is our simpler equation, we'll substitute this expression for y into the second equation.Notice how we replace y with the expression x plus 2, keeping the parentheses to maintain order of operations.Now we can distribute and combine like terms. The x squared remains, and we bring down x plus 2.Finally, we subtract 5 from both sides to get our standard form quadratic equation: x squared plus x minus 3 equals zero.Let's look at how we combined these terms: The x squared term stays as is, we keep the x term, and the constant terms combine.This is our final quadratic equation in standard form, ready to be solved in the next step.Now that we have our quadratic equation, let's solve it by factoring.Setting each factor equal to zero gives us our x values: negative two and positive one.Let's find the corresponding y-values by substituting these x values into our linear equation.For x equals negative two, y equals zero.For x equals one, y equals three.It's crucial to verify these solutions by plugging them back into both original equations.Let's verify the point negative two, zero. It satisfies both the linear and quadratic equations.Similarly, the point one, three also satisfies both equations.Let's review the key points about solving mixed systems of equations.Thanks for learning about solving mixed systems with Spark.E!