3.1 Linear equations
We encounter many situations in which we have to solve
linear equations (that is,
Consider the problem: What is the number x which when multiplied by 8 gives 125?
We know at once that the number x must be greater than 10
since 10 times 8 is 80 but less
x(8) = 125 ___ or ___
Since 8 times x is to be 125, we reason that x must be the
number that is obtained by
So, the number is ___ or ___ . The number ___ or ___ is
called the solution of
To solve more complicated problems, we make use of the
following properties of
(i) If equals are added to equals, the results are equal.
Since subtraction of a number is defined to be the
addition of its negative, the statement,
We can formally solve the simple equation given above as follows:
Of course, we do not write in such great detail, but it is
important to know the principles
Example 1: Find the solution of the equation
Solution: We will write out the solution in great detail.
Once you get used to, you
The idea is to bring the equation to the form ax = b by
making use of the
To get rid of ___ from the right-hand side, we add ___ to
both sides of
Example 2: Find the solution of the equation 5(3t+ 2) = 2(5t+ 7)+ 8 .
Solution: Here the first problem is to simplify both sides of the equation:
Therefore, the solution of the equation is 2.4.
Example 3: Find the solution of the equation
Solution: Here again the first step is to simplify the
left-hand side of the equation.
The solution is 3.55.
1. Find the solution of each of the following equations:
(You should check your
3.2 Literal equations
We often have to solve an equation for one of the
variables in terms of the other
Example 1: A man wants to buy a new car priced at
$18,000, financing it at 5.75%
We could have substituted the values into the installment purchase formula
and obtained the equation
We can solve this equation as follows:
So, the monthly payment is $345.90.
Or we can substitute the values in the monthly payment formula
which is exactly the same as the value obtained above.
Let us review how we obtained the monthly payment formula
from the installment
Then, we have
We have to solve this equation for M. Adding to both sides of the equation, we get
Multiplying both sides of the equation by C and then
dividing both sides of the equation
Substituting back, we get the monthly payment formula:
Solve each of the equations for the indicated variable: