We continue to study system 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