INTERMEDIATE ALGEBRA - CH 4, LECTURE
4.1 Systems of Linear Equations in Two Variables
Checking solutions - by sub
Check one example
Graphing method:
Ex: y=3/5 x - 3, 2x - y = -4 Solution (-5, -6)
Pay close attention to identify all grid point solutions in the vicinity of the intersection
Types of solutions The key: the solutions are the intersections
Crossed lines - (The usual result) One intersection
One solution (one ordered pair)
Parallel lines - Example: one equation is a multiple of the other, except for the constant.
No solution (they never intersect)
How to recognize: 0 = 4 (variables drop out; we get different numbers: result always false)
Infinite solutions - Make up ex - one equation is a multiple of the other, including the const
Same line
How to recognize: e.g. 0 = 0 (variables drop out; get same number; result true regardless of x)
Substitution method
Most convenient when one equation is already solved or easily solvable for one variable
Previous text:
This technique is usually the technique of choice for nonlinear systems
Ex: Solve system {y=x2-6x+8 , x-y=2}-- sol is (5,3) or (2,0)
Application:
Break-even point
Mixture problems using x and y
Investment problems
Number problems or geometry problems
Previous textbook: Special case
Example with terms of the form a/x and b/y - Replace 1/x by u, replace 1/y by v
4.2
Elimination /Addition method
Most convenient when both equations are of the form ax + by = c.
Make up random example with fractional coeffs, clear fractions first
Std technique: back-sub to get second var
Alternate technique: just elim other var in order to get second solution
(apply to the same example)
Important key: clear fractions before doing the elimination.
Applications: Same types as 4.1
4.3 Systems of Equations in Three Variables
Solution technique
Pick one var to elim
Elim that var from two pairs of equations
Result is a smaller system of eqns (one less eqn, with one less var)
Repeat with the reduced system
Unique solutions - normal
Ex: clear fractions
No-solution case
No unique solution (inf solution) case
4.4 Matrix
solutions: Gaussian Elimination
Work toward row-echelon form
If you can get the leading coefficient to be 1, that's best
If not, you can still get the solution to the system
4.5 Determinants
Def and Calc technique for 2X2
Example for solving a
det eqn for x: 
Calc tecniques for 3X3
"Shortcut" technique
Expansion by minors - if possible, pick a row/column with zero(s)
Sum of (alternating sign X coeff X minor)
Application: Equation for line through 2 points (x1,y1), (x2,y2)
Expand along the top row to get the general form of the line (or a multiple):
Ex: 
, Expand: (7-4)x + (3+2)y + (12+14)=0. Ans: 3x + 5y = -26
Same task using 2´2 determinants: Find eqn of a line given 2 points, organized in an x-y T-table
Calculate delta x and delta y by subtracting top minus bottom
Calculate D = determinant of the T-table; then:
m = delta y / delta x, and
x y 3 7 -2 4
b = D / delta x
Ex: for we have delta x = 3 - (-2) = 5
delta y = 7 - 4 = 3
D = 12 - (-14) = 26
therefore, y = 3/5 x + 26/5
Cramer's Rule
See the theorem: Prove calculations for D and Dx: Solve syst for x by eliminating terms with y
State full thm for 2 vars
Note how to adjust D to get Dx and Dy.
Computational example: (random numerical coeffs)
Extend to 3 vars
Computational example:
App: discovering whether a unique solution exists: No unique sol. if D = 0.
Form constant/0 for any var -- No solution
Form 0/0 for all vars -- Dependent system (inf solutions)
Solving a syst where the coeffs contain vars
Ex: 2x2
McKeague
3.5 Applications
Steps:
1. Read and list facts; (build and label a table, graph or picture)
2. Assign variables and translate information
3. Write a system of equations
4. Solve the system
5. Write answers
6. Reread and check
Types (from exercises) -- Show setup only on these in class.
Number [Example 1] In class, set up Exercise 6.
Ticket and interest [Example 2] In class, set up Exercise 8.
Mixture [Example 4] In class, set up Exercise 16 (classic chemical)
In class, set up Exercise 19 (mixture with costs and quantities)
Rate [Example 5] In class, set up Exercise 22 (upstream/downstream)
Coin [Example 6] In class, set up Exercise 28: consider value per coin
Linear modeling [Example 7] In class, set up Exercise 33 Recall technique described in 3.3
Curve fitting In class, set up Exercise 40. Subst into general eqn of the curve to get a system of eqns
3.6 Systems of Linear Inequalities
Just like indiv linear inequalities, but
shade region that make all the inequalities true
Inverse problem: given the graph, write the syst of inequalities.
For apps, we have to write the inequalities, and then graph: X27
Appendix B
What I really want you to get from this lesson:
Importance of row echelon form: If you have this form, you can solve the entire system by back-substitution.
Elementary Row Operations: How to do #3
Use of a pivot to zero out all entri
Elementary Row Operations (the "how to do each step")
Interchange any two rows (swap rows) -- these first two operations should be familiar
Multiply or divide any row by a [nonzero] constant
Add a multiple of any row to any other row -- this is new for Math C; learn how to do it
Strategy (the "what to do next")
If possible, reduce the smallest entry in the first column by adding a multiple of one row to another
Swap rows to get the smallest entry into the upper left position (the "pivot" - nonstandard term)
Multiply or Divide: Either
Divide first row by the pivot number (which will make the pivot = 1), or
Multiply the other rows so that their leading entries ("targets") are multiples of the pivot
Get zeros below the pivot by adding to each row the appropriate multiple of the pivot row
divide the target element by the pivot and reverse the sign
[I like to get zeros above the pivot the same way]
Repeat the process for the smaller matrix below and to the right of the previous pivot.
When finished, divide each row by the leading entry in each row (so each row begins with 1).
Examples:
2´2: #16
3´3: #10
Nasty example with all the bells and whistles:
Mc Keague
4.6 Equations Involving Rational Expressions
Clearing fractions. 3 Techniq ues
1. General procedure: multiply both sides of eqn (therefore every term) by the LCD
2. For proportions, clear fractions by cross-multiplying
Example: 
Solve 
for y:
Cross-multiply to get: x (y − 1) = y−3
Clear parentheses: xy − x = y − 3
Collect terms with y on the left: xy − y = x − 3
Factor out y: y (x − 1) = x − 3
Divide to finish: y = (x − 3 / (x − 1)
Exception: Use the general procedure if the denominators have a common factor
3. For proportions, it is sometimes faster to take the reciprocal of both sides
Example: Solve 
The solution is x =
3/7.
[but you can't take reciprocals term by term, where either side has two or more terms]
Ex. To use this technique on 
first add the
fractions on the right: 
;
then take
reciprocals: 
4.7 Applications
Number problems
X2
Key: write "1/x" for "the reciprocal of x"
Motion problems
Upstream-downstream problems with same time: solve both eqns for t, set results equal
Work problems:
Std setup 1/a + 1/b = 1/t
We are really adding rates: how much of the job gets done per unit of time?
For example 4 in book: easier to consider the net increase per hour, 1/10 - 1/12 =1/60 pool/hour
Unit analysis
X31 ft/sec to mph
X33 rpm to mph [what is the distance for one revolution? C = 2pr]
Graphing rational functions of the form (ax+b) / (cx+d)
1. Plot y-int [it will be the ratio b/d]
2. Plot x-int: solve for num = 0 [we need the entire fraction = 0, but the denom is irrelevant]
3. Plot the vertical asymptote [def: line the function approaches at the ends of the graph]
solve denom = 0
This is the value the fraction can never have
as x gets close to this value, y becomes very large (in absolute value)
positive on one side, negative on the other
4. Plot the horizontal asymptote
It is the ratio a/c
Explanation: as x becomes very large, the constants (b and d) become relatively insignificant
The function becomes close to ax / cx, which reduces to a/c
5. Plot at least one point on each side of the vertical asymptote
(The y-intercept may serve as one of these points)
6. Sketch the graph through the points and approaching (but not touching) the asymptotes.
Ex: y = (4x+9) / (2x - 3)
y-int at -3; x-int at -9/4 = -2.25; VA at x = 3/2 = 1.5; HA at y = 4/2 = 2; extra point (3,7)
Return to: Merced College; Don Power Updated 10/08/07 by Don Power