Linear equations

Linear equations

Any time we write something with an equal sign, we write an equation.

• Variable - Any quantity in an equation which we do not know is called a variable.

• Constant - Anything in an equation which can take on only one value is called a constant.

• Term - The product of a constant and a variable raised to some power is called a term.

• Coefficient - The constant in such product is called a coefficient

 $3x+y=2$  

• x and y are variables

• 2 is a constant term

• 3 is the coefficient of x

The degree of a term is the sum of the powers of the variables in that term. For example, the dgree of x is one, and the degree of xy² is three. The degree of an equation is the highest degree of the terms in the equation. Linear equation is an equation whose degree in one.

• Is  $x+\sqrt{x}=3$  a linear equation?

• How about  $xy=4$  ?

One Equation, One Variable

Two Equations, Two Variables

• One equation, add/abstract then divide/multiple

• Two equations: Substitution or Elimination

Solve equation
 $2x-4y=7$  

 $x-2y=2$  

• By using elimination, we get 0 = 3

• never be true. no solution.

Word Problems

• Define all the variables

• Write the equations described by the problem

• Solve the equations

Johnny is twice as old as Gina. Johnny is five years older than Gina. How old is Johnny?

• Step 1: variables - Johnny's age is x, Gina's age is y.

• Step 2: x = 2y, x = y + 5

• Step 3: Solve it.

Common class of problems

• "Moving" problem - rate x time = distance

• "Job/Project" problem - (time at work)(fraction of work in one unit of time) = (fraction of job done)

"Moving" problem is very common. Following is an example of "Project" problem.

Pipe A can fill a pool in 5 hours, while pipe B can fill it in 4 hours. How long will it take for the two to fill the pool if both are operation at the same time?

• In one hour, pipe A can fill 1/5 of the pool. pipe B can fill 1/4.

• In one hour, both of them fill (1/5+1/4)

• For x hours, they fill x(1/5+1/4). ( when this fraction is 1, the pool is full)

Another 'Job' problem.

Tom and Huck paint a fence for four hours, after which Jim helps them and they finish two hours later. If Jim had not helped them, it would have taken them 5 more hours to paint the fence. How long will it take for Jim to paint the fence alone?

• Variables - Tom x, Huck y, Jim z

• Equation - 4(1/x+1/y)+2(1/x+1/y+1/z) = 1

• 4(1/x + 1/y) +5(1/x+1/y) =1