Unit 5 Systems of Equations & Inequalities is an important topic in Algebra. It deals with solving a set of equations that contain two or more variables. In this unit, we will learn how to solve systems of equations and inequalities using various methods.
Solving Systems of Equations
One of the methods to solve systems of equations is the Substitution Method. In this method, we substitute one equation into another to eliminate one variable. For example, if we have the equations 2x + y = 5 and x – y = 1, we can solve for y in the second equation and substitute it into the first equation. This gives us 2x + (x – 1) = 5 which we can simplify to get x = 2. We can then substitute x = 2 into either equation to solve for y.
Another method to solve systems of equations is the Elimination Method. In this method, we add or subtract equations to eliminate one variable. For example, if we have the equations 3x + 4y = 10 and 2x – 3y = 5, we can multiply the second equation by 2 to get 4x – 6y = 10. We can then add this equation to the first equation to eliminate y and get 7x = 20. We can then solve for x and substitute it into either equation to solve for y.
Solving Systems of Inequalities
Systems of Inequalities involve solving two or more inequalities at the same time. For example, if we have the inequalities x + y >= 3 and x – y <= 1, we can graph each inequality on the same coordinate plane to find the solution set. The solution set is the region shaded by both inequalities.
We can also solve Systems of Inequalities algebraically by using the same methods we use to solve Systems of Equations. We can use the Substitution or Elimination methods to eliminate one variable and then solve for the remaining variable. We can then plug in the value of the remaining variable into the original inequalities to find the solution set.
Applications of Systems of Equations & Inequalities
Systems of Equations & Inequalities have many real-world applications. One common application is in economics where we use Systems of Equations to model supply and demand. Another application is in engineering where we use Systems of Equations to model the behavior of electrical circuits or mechanical systems. Systems of Inequalities are also used in optimization problems where we want to maximize or minimize a certain objective function subject to certain constraints.
Conclusion
Unit 5 Systems of Equations & Inequalities is an important topic in Algebra. We have learned how to solve Systems of Equations using the Substitution and Elimination methods, as well as how to graph and solve Systems of Inequalities. We have also seen some real-world applications of these concepts in economics, engineering, and optimization problems.