Name: Mrs. Gorsline Integrated Math 2Unit 5 Guided Notes

Hour:

5-3 Homework Notes

5-3: Solving Systems by Substitution

Review: A system of equations is when you have multiple functions graphed on the same coordinate plane. The solution to the system of equations is where the graphs ______, or cross.

To find the exact solutions to a system of equations, you have to use ______techniques. The first technique that you will learn uses the ______Property of Equality, which states that if ______, then a may be substituted for b in any arithmetic or algebraic ______.

Example:

Solve the system by substitution.

5-3 In-Class Notes

Substitution can also be used when there are more than two ______and two ______.

Example: Suppose a part of a stadium has a seating capacity of 4216. There are four times as many lower-level seats as there are upper-level seats. Also, there are three times as many mezzanine seats as there are upper-level seats. How many seats of each type are there?

Not all graphs within a system of equations will be ______.

Example: Solve the system

These examples illustrate that ______may be an appropriate method when at least one of the equations has been or can easily be solved for ______of the variables.

Systems are classified into two groups depending on whether or not ______exist. A system which has ______solutions is called a ______system. A system which has ____ solutions is called an ______system.

Example: Solve the system

HW: Complete p. 289 # 2,3,5,6,7 on a separate sheet.

5-4 Homework Notes

5-4 Solving Systems Using Linear Combination

When linear equations are written in ______form, it may be more efficient to solve the system using the ______and ______Properties of Equality rather than Substitution. This method is known as the ______or Elimination method.

Example: Solve the System by adding the two rows together and eliminating a variable.

Now use substitution to find the second variable to get your final solution.

5-4 In-Class Notes

Example: The table below shows the protein and calcium contents for a dinner of roast beef and mashed potatoes. How many servings of each are needed to get 29 grams of protein and 61 milligrams of calcium?

Roast Beef / Mashed Potatoes
Protein (g) per serving / 25 / 2
Calcium (mg) per serving / 11 / 25

Realize that when using this method, there are often ______ways to solve the system and get to the same ______. I would suggest doing the method that will give you ______numbers.

This method can also be used to solve a system with ______linear equations. First, take any ______of equations and eliminate ______of the variables. Then take ______pair and eliminate the ______variable. This will leave you with ______equations with ______variables each. Then ______another variable by ______those two equations. Finally, use ______to go back and solve for the other two variables.

Example: Solve the system

HW: Complete p.295 #2,6,7,15,16on a separate sheet.

5-6 Homework Notes

5-6: Solving Systems using matrices

In the middle of the nineteenth century, the British mathematician ______developed a way to solve systems of linear equations using ______. Notice that we can represent the following system:

by the matrix equation

(complete this matrix multiplication below if you cannot see how this works)

This is called the ______. The matrix is called the ______matrix, the matrix is called the ______matrix.

If you were to solve the equation , you would have to multiply both sides of the equation with the inverse of 3, which is . You must do the same thing to solve the matrix equation above.

Example: Solve the equation from above for x and y by multiplying both sides of the equation by the inverse of the coefficient matrix (see p. 306 for help).

Complete p. 310, #2-4

5-6 In-Class Notes

In general, to solve the system using matrices, first rewrite the system as a matrix equation:

To solve this equation, multiply each side by ______.

Therefore, the solution to the system is always going to be the inverse of the ______matrix times the ______matrix. Make sure, however, that you always put the coefficient matrix ______, otherwise you can not multiply because the ______won’t match correctly.

Example: Ku invested $25,000, some in a savings account and the rest in bonds. If the return on his savings account is 4% and the return on his bonds is 6%, how did he divvy up his investments if he earned a total of $1300 in interest?

Notice that you can’t solve a system using matrices if the ______of the coefficient matrix doesn’t exist. Remember that this happens when the ______equals _____. This means that as long as the determinant ______equal ___, then the 2x2 system will have ______solution.

Matrices are the best way to solve a system of equations that have ______variables, because they can easily be solved in your ______.

Example: Solve the system using matrices and your calculator.

Complete p. 310-311, #6-9,11 on a separate sheet

5-7 Homework Notes

5-7: Graphing Inequalities in the Coordinate Plane

Inequalities with one variable can be graphed either on a ______or in the ______.

Example: Graph the solutions to using both of the above methods.

Remember that when there is no equal sign, then your line must be ______.

Remember that if the equation is some number, the line will be drawn ______, where if the equation is some number, the line will be drawn ______.

Example: Graph the linear inequality

Complete p. 316 #4-6

5-7 In-Class Notes

What if you have an inequality story problem, where your values can only be ______? If that’s the case, then you can’t just ______, you will have to use ______.

Example: A ferryboat transports cars and buses across a river. It has space for 12 cars, and a bus takes up the space of 3 cars. Draw a graph showing all possible combinations of cars and buses that can be taken in one crossing.

Graph some more challenging inequalities below if time.

Complete p. 316 #9,10,15,16,19,20 on a separate sheet

5-8 Homework Notes

5-8: Systems of Linear Inequalities

The set of solutions to a system of linear inequalities is often called the ______or ______for that system. The ______are always parts of lines. The ______of the boundaries are called ______of the feasible set.

Warning: If you want to shade both inequalities and then use just the overlapping part as your solution, you MUST either use different colored pencils or shade very lightly at first, and then shade the final answer much darker.

Example: Graph the feasible set for the system

5-8 In-Class Notes

The applications of ______to production planning in ______may involve systems with thousands of variables. Computers are needed to solve these problems (and some engineers do this for a living!). However, simple examples can be done by hand.

Example: The Biltrite Furniture Company makes wooden desks and chairs. Carpenters and finishers work on each item. On average, the carpenters spend four hours working on each chair and eight hours on each desk. There are enough carpenters for up to 8000 worker-hours per week. The finishers spend about two hours on each chair and one hour on each desk. There are enough finishers for a maximum of 1300 worker-hours per week. Given these constraints, find the feasible region for the number of chairs and desks that can be made per week.

Start by making inequalities that represent all of the constraints given to you.

HW: p. 322 #3,4,5,11,12

5-9 & 5-10 Homework Notes

Remember the problem about the chairs and the desks. We figured out the following equations. We are using x for chairs and y for tables.

Let’s say that we were told that the company earns a profit of $15 per chair and $20 per desk. How can the production schedule be set to maximize the profit? The greatest profit will occur when the line with the equation ______is as high as possible, but still intersects the ______. This must happen at one of the ______.

This type of problem is called a ______problem.

Complete the worksheet.

HW: Complete. P. 329 #13-15