UNIT 2 ALGEBRA
This chapter covers the algebraic
operations most used in Quality Control calculations. Relationships among
engineering variables (length, weight, velocity, etc. )
are normally expressed in algebraic equations. The
relationship among the length, width, and area of a
rectangle is that the area is equal to the length times
the width. This is expressed algebraically by the
equation, A = L x W. To get the area of a rectangle we
multiply the length by the width. If, however, we knew
the area we wanted a rectangle to have and knew the
width, we would have to calculate the length required. To
do this we would use the rules of algebra and change the
equation, A = L x W to
. We need
to get the unknown variable by itself on one side of the
equal sign (normally the left side) and all the other
variables on the other side of the equal sign.
After completing this chapter the student will be able to:
- use standard notation in writing algebraic equations.
- solve straight line equations for any variable.
- solve simple curve equations for any variable.
For demonstration purposes an "x" is often used to show multiplication, as in the equation A = L x W. Where the "x" may be confused for another variable, a "*" is used, for EXAMPLE A = L * W. Normally the multiplication symbol is left out giving A = LW.
In some relationships, the same factor may appear more than once. In EXAMPLE 2 of Chapter 1 three volumes were calculated. To FIND the volume of the steel container the volume of the empty space was subtracted from the outside volume of the container. In order to keep the letters V straight they were subscripted, Vsteel = Voutside - Vinside. If the volume were written "VSTEEL" it may have been interpreted to mean V * S * T * E * E * L.
When the same variable is multiplied by itself, for EXAMPLE A * A, this is expressed by an exponent. A * A = A2. A * A * A = A3. The area of a circle is expressed as A = p r2 , not A = p rr.
When the same variable is to be multiplied by two variables which are added or subtracted, for EXAMPLE, B H1 + B H2, this can be expressed by including those variables in parenthesis and placing the common multiplier outside the parenthesis, B (H1 + H2 ). In order to solve an equation for one of the variables inside the parenthesis, they must be removed by multiplying each variable inside by the outside multiplier.
A( b - d ) becomes Ab - Ad.
Six Rules for Preservation of an Equality
In order to solve an equation for one of the variables, that is get one of the variables isolated on one side of the equal sign, we have six rules we may apply.
2. Divide both sides by the same quantity.
GIVEN: A = L * W
FIND: L
A = L * W
ANSWER:
3. Add the same quantity to both sides.
GIVEN: D - F = G
FIND: D
D - F = G
D - F + F = G + F
ANSWER: D = G + F
4. Subtract the same quantity from both sides.
GIVEN: D = G + F
FIND: G
D = G + F
D - F = G + F - F
D - F = G
ANSWER: G = D - F
5. Raise both sides to
the same power.
6. Extract the same root
from both sides.
EXAMPLE 1 
1 GIVEN:
A = X + 2Y
FIND:
a. X = ? (Rule 4)
b. Y = ? (Rule 4, Rule 2)
There are several short cuts to applying some of the six rules
1. Move a variable across the equal sign and change the sign.
EXAMPLE 2
Short cut: Move the "F" to the left of the equal sign and change its sign.
ANSWER: D - F = G
2. Cross multiply diagonally across the equal sign.
EXAMPLE 3
GIVEN: A =
FIND: H
Rule: Multiply both sides by the same
quantity.
Short cut: Cross multiply diagonally across the equal sign.
ANSWER: H = 2A
3. Move a multiplier diagonally across the equal sign.
EXAMPLE 4
Short cut: Move the multiplier W diagonally across the equal sign.
A = LW
EXAMPLE 5
In this section, the rules of algebra will be applied to solve geometric problems.
| A rectangle must have an area of 16 square inches. The length dimension must be 3.5 inches. What must be the width? |
EXAMPLE 6
 |
EXAMPLE 8
EXAMPLE 9
GIVEN:
| A = 6 in2 H1 = 1 in B = 3 in |
 |
EXAMPLE 10
 |
EXAMPLE 11
| A can is to hold two gallons. If the height must be 12 in, what must be the radius? |
EXAMPLE 12
The volume of a
sphere is GIVEN by the equation  What
must be the radius of a sphere with a volume of
231 in3. |
These problems provide an opportunity to demonstrate and reinforce skills in solving technical problems. Solve them using the examples as a guide. When done contact your instructor for solutions and a self-grading guide. Do this problem using the GIVEN, FIND, ANSWER format. Include a dimensioned sketch as part of the GIVEN.
