Question

Question: Describe the various methods of solving linear systems. With which method of solving linear systems are you most comfortable, and why?

Hint: First, define a linear system, and give an example. Then, discuss the methods, and show the steps to solve your example. Finally, talk about advantages and drawbacks of each method.

"Real-Life" Relationship: Any relationship where we have a fixed cost and variable cost can be represented by a linear equation.

For instance, the cost of a rental car from Hertz might be $100 plus $0.70 per mile, while Enterprise might charge $80 plus $0.80 per mile. We can solve the following system to find out when the cost is the same (c = cost, m = miles driven)

c = 0.7m + 100 (Hertz)

c = 0.8m + 80 (Enterprise)

It turns out that they are equal when the mileage is m = 200.

Challenge 1: Given two lines in standard form, how can you quickly decide if they have the same slope, simply by using ratios of y and x coefficients?

Challenge 2: What is a quick way (without finding the slope or solving for y) to decide whether the following system has a solution?

2x + 3y = 4

2x + 3y = 5

Answer 1

#question

A system of linear equations is just a collective group  of two or more linear equations.

for example:-

2x+3y = 1 ......(1)

x+y = 2 .......(2)

equation(1) and equation(2) creats a system of two linear equations having two variable x and y.

there are basically two methods

(a) substitution method:- in this method we solve one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging this back into the other equation, "substituting" for the chosen variable and solving for the other. Then you back-solve for the first variable.

x+2y = 3 .....(1)

x+y = 1 .....(2)

by equation(2)

y = 1-x .....(3)

substitute the value of y in equation(1)

x+2(1-x) = 3

-x+2 = 3

x = -1 and y = 2

(b)elimination method:-

In the elimination method you either add or subtract the equations to get an equation in one variable.When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.

x+2y = 3 .....(1)

x+y = 1 .....(2)

subtract equation (2) from equation(1) . we get

y = 2

now put this value of y into equation(2)

x = -1

both methods are almost simillar .in my experience there is no advantage or disadvantage of one method over other.

#challenge-1

find the ratio of respective coefficients of x and y , if they are same then slope is same otherwise not

#challenge-2

2x+3y = 4 ....(1)

2x+3y = 5 ....(2)

by equation (1) and (2) ,

4 = 5

which is not true means this system have no solution.