We continue to study systems of linear equations, with the focus on the word problems involving systems of linear equations.

Such work problem usually can be modeled with two variables x and y, and the given conditions should allow you to come up with two equations using x and y. Then you solve the equations to come up with the solution. Be careful to check if the solution makes sense for the real life.

We look at several challenge problems where by transformation, we can solve them by systems of linear equations.

Homework:

Page 55: #34 – #48

We study system of linear equations. Here we talk about linear equations with two unknown. So a system of linear equations is with two linear equations. The solution of x and y that satisfies both equation is called the solution to the system.

To solve it in algebraic way, there are basically two methods:

With “Elimination Method”, we try to multiply some numbers to one or both equations, so that by adding or subtracting the two equations, one unknown is gone. Once we solve the first unknown, we put in back into one of the original equation to solve the other.

In the simple elimination case, you can just add or subtract the two given equations. Often you need to pick an unknown first, look at the coefficients of that unknown in both equations and come up with the LCM. Then multiply to each equation some number so that the new equations have that unknown with the same coefficient. Now you can do add or subtract.

With “Substitution Method”, you start with one given equation, rewrite the equation using one unknown to represent the other. Now substitute this unknown in the second equation with what you have, so now you have an equation with only one unknown.

We look at the special case where there is no solution to a system of linear equations, which shows two parallel lines in you graph them, and the special case where there can be infinite number of solutions, basically the two equations are identical. In the graph, you have the same line.

Here is this week’s homework:

Page 53, #1 – #12; Page 54, #21 – #25, #31

More graphing and graphs.

We first examined travel graphs. A travel graph is a graph showing the relationship between the distance traveled and the time taken. With a travel graph, we can calculate the average speed (total distance by total time), a speed for a particular period, measuring the total distance traveled, etc. When a travel graph has information about two objects (like two cars), we can further see when a faster catches a slower one, or when two cars meet, etc.

We then looked at solving system of linear equations by graphs. Here we talk about two linear equations with x and y variables and find a solution which satisfies both equations. On the graph, it should be the intersection point of the two lines representing the two linear equations. Of course, there are some special cases where there is no solution due to the two lines being parallel to each other, or there are infinite number of solutions when two lines are the same.

Lastly we examined the graphs for a quadratic equation, taking the form of y = ax^2 + bx + c. Here we just need to student to have a basic knowledge of the shape of a quadratic function, its symmetry axis, the existence of a maximum or minimum point.

Here is this week’s homework:

Page 40: #6, #7, #8, #9, #21, #22, #23, #24, #34, #35, #36

Today we went over problems involving financial transactions first.

Here we talk about profit (selling price – cost) and profit percentage (profit / cost), discounted price (usual price – discount percentage * usual price), interests. This is mainly the continuation of what we learned last week, about rate, ratio and percentage.

We move on to coordinate plane and representing ordered pairs in the coordinate plane, drawing graphs for linear equation with two variables, and how to use linear graph to solve simple problems.

Homework:

Page 26, #22 – #28

Page 38, #3, #4, #5, #16

For midterm exam results, I will communicate with each family individually this week.

Today we first cover the section for word problems leading to quadratic equations, question like find the number such that the sum of the number and its square is 156. We go from the given conditions to come up with a quadratic equation, solve it. With real world problem, we need to be care to pick the right answer as a quadratic equation can have two solutions, one is possible, the other can be impossible in the real world.

We move on to talk about rate, ratio and percentage, and the corresponding word problems.

Rate is used to describe how a quantity change with regard to another quantity. For example, a worker is paid $84 for 6 hours of work, we say that he is paid $14 for one hour, or at the rate of $14/h.

Ratio is used to compare the magnitude of two similar quantities. It indicates what fraction one quantity is of the other, or how many times one quantity is as much as the other. Here is an example: a sum of money is divided between X and Y in the ratio of 3:5. if X gets $x and Y gets $y, then we can write: x:y = 3:5, or x/y = 3/5, etc.

Percentage is a special fraction with one hundred as its denominator. For example, 15% = 15/100 = 0.15

There are quite some word problems involving rate, ratio, and percentage, and some of them can be challenging,. We talk about several examples in the classroom.

Homework:

Page 21: #27 – #32

Page 24: #2, #4, #5, #9, #12, #14, #18, #19, #21, #25

Today is the review day and we went over what we have learned this semester in the first 6 lessons: Indices (Exponents); Algebraic manipulations; Literal and quadratic equations, and factorization of quadratic expressions.

We will have our midterm exam next week on 11/3. Please go over your homework. To help you prepare, you can also do the following on your workbook:

Page 11, Test Paper 1, question 1 through 8, and Page 32, Test Paper 2, question 1.

Today we mainly focused on quadratic expression factorization (因式分解). This is a skill that is very useful in quickly solving quadratic equations. While it is not hard, it does take practice to get good at. So we spent most of the time letting the students going exercises.

Basically, for the simple form of x^2 + bx + c (that is, the coefficient in front of x^2 is 1) to be able to be factored as (x + p) (x + q), then p + q must be equal to b and pq (p times q) must be equal to c. So the basic steps are, first look at the factors for c and find two whose product is c, then check if their sum equals to b, if so, you have it. If not, try different pairs.

Then for the general form of ax^2 + bx + c (a <> 0), it is the same idea but a little more complicated. For it to be factored into (px + m)(qx + n), then pq (p times q) must be equal to a, mn( m times n) must be equal to c and pn + qm must be equal to b. The basic steps are, first look at the factors for a and pick two so that its product is a, then pick two factors of c so that its product is c, then check if the cross multiplication of those two pairs is b.

A cross multiplication chart will help

px         +m        | +pnx

X             |

qx          +n        | + qmx

—————————————-

pqx^2 + mn   | (pn + qm)x

With this skill, we looked at solving quadratic equations where we factor first.

Note, we have not got into the general solution of a quadratic equation yet.

Homework:

Page 19 18 a c e g i k m  o, 19 b d f h j l n, 20 a ii, f ii, g ii, i ii, j ii, k ii

We talked fractional equations and word problems with fractional equations.

Any equation involving any fraction, either x (unknown)  in enumerator or denominator, is called a fractional equation, or also called rational equation.

We studied fractional equations with x in enumerators, and we studied equations with x in denominators. When x is in denominators, one important step is to check the answer to make sure that it does not make any denominator zero and throw away any answer that does.

For word problems, it is important to come up with the right equation with the given information.

Homework is as follows:

Page 9: 49 a c d e g i k m n o, 52 a, 53, 54, 55, 56