First assignment: For next class, make a list of any questions you
have regarding the syllabus. If you have no questions, bring in a
sheet of paper that says "I understand and agree to the syllabus
for MTH 103.202," sign it, and date it. This is worth 5 points.
There is one last, and very important, step: checking the answer.
Our solution is , so we plug this in for t:
which it just so happens is a true statement. So our
solving linear equations - 3
To recap, solving a linear equation involves four steps:
1. Simplify to get a sum of terms on each side
2. Collect constants on one side and variables on the other
3. Combine each side\'s terms
4. Multiply or divide to get variable = constant
and the additional step of checking the answer.
In fact this same strategy can be used to solve equations which
aren\'t linear, but resemble linear equations.
in which variables appear in the denominator.
We have to be careful, though. In this particular example, what
happens if we try the value x = 0?
We get. But division by zero is undefined, so this equation doesn\'t make
sense! ((We\'ll talk about why next class.))
So at the very beginning of the problem, we have to make sure we
aren\'t diving by zero. Here that means we can\'t let x = 0.
solving equations with variable denominators - 2
This is a linear equation!
and we notice that our requirement x ≠ 0 is satisfied by
solving equations with variable denominators - 3
But our restriction is x ≠ 3, so this solution is invalid.
equation is inconsistent.
We write that the solution is Ø ?.
What good is all of this? Well, it turns out plenty of things in life
can be modeled by linear equations.
A lot of them have to do with money.
applications - example
I\'m going to pay you to do your homework. I\'ll pay 8
cents for each problem you get right, but you owe me 5
cents for each problem you get wrong. You do 26
problems, hand in your work, and I tell you we\'re even.
How many problems did you get right?
First pick something to be the variable. In this case, let\'s take c to
be the number of problems you got correct.
Then write the information given in terms of that variable.
The total number of answers is 26, so the number of incorrect
answers is 26 - c.
Now we can write down an equation.
money from right ans. minus money from wrong ans. is zero
applications - example - 2 So we\'re left with the equation
We\'ll come across the problem of interest in several guises.
Definition The simple interest earned on an investment of P, at the annual
rate r , for t years, is I = Prt
You have $1000. You invest part of it at a simple interest
rate of 5%, the rest at a simple interest rate of 8%. At
the end of one year, you have earned $56 in interest.
How much did you invest in each account?
Pick a variable. x is the amount invested at 5%
Then 1000 - x is the amount invested at 8%
interest at 5% plus interest at 8%
applications - interest - 2
So we need to solve
The question was "How much did you invest in each account?", so
we need to answer accordingly. $800 was invested at 5%; $200 was invested at 8%. Does the answer make sense in the context of the original
problem? Does the math check ?
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