Download Section 3.1

Review for Final
Systems of Linear Equations
Two Equations, Two Unknowns
• Solve;
– Graphically
– By substitution
– By addition (elimination)
• When there is no solution
• When the lines are identical
What is a solution?
• A solution is a coordinate pair that satisfies both equations
• 2x – 3y = 3 and 4x – 3y = 9
• The coordinate pair (3,1) satisfies both solutions
• To verify this, substitute the pair:
2(3) – 3(1) = 6 – 3 = 3: OK
4(3) – 3(1) = 12 – 3 = 9: OK
Solving by Graphing
• This is quite hard to do well – you have to be very accurate
when drawing your graph
• The solution to a system of equations must satisfy both
equations; that happens when the lines cross.
Graphing a Line
• Need a minimum of two points, three for safety
• Can use:
a table of points generated from the equation
the intercepts
a point and the slope
• To verify if a point is on a line, substituted the point into the
equation and see if it works
Find the solution
• -x + 3y = 10 and x + y = 2
8
6
4
2
0
-8
-6
-4
-2 -2 0
2
4
6
-4
-6
-8
• The answer seems to be (-1, 3); check
-(1) + 3(3) = 10 and (1) + (3) = 2; it works!
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Solving by Substitution
• x + 2y = 10 and 3x + 4y = 8
• Get one equation in the form of x = qqqqqq or y = qqqqqqq
• Substitute for x or y in the other equation and solve
• Here, the first is easy: x + 2y = 10 becomes x = 10 – 2y
• Substituted for x in the other equation:
3(10 – 2y) + 4y = 8
30 – 6y + 4y = 8
-2y = - 22, y = 11
• Find x: x = 10 – 2y = 10 – 22 = -12
• Solution is (-12, 11); substitute back in to check
Solve by Addition (Elimination)
2x + 3y = 9
4x + y = 8
• Multiply the first equation by -2 and add:
-4x -6y = -18
4x + y = 8
-5y = -10 or y = 2
• Substitute to find x: 2x + 3y = 9 becomes 2x + 6 = 9 or x = 3/2
• Check solution: 4(3/2) + 2 = 6 + 2 = 8; correct
2(3/2) + 3(2) = 3 + 6 = 9; correct
Fractions?
• Since it is an equation, can eliminate, if choose:
• x/2 + y/6 = 2/3 becomes
• 3x + y = 4
Inconsistent Equations
• Inconsistent equations have no solution
• Graphically, the lines are parallel
• If you can verify the slopes are the same, you can save
yourself some time
• Else: y = 3x – 6 and 6x – 2y = 9
substitute y = 3x – 6 into the second, getting:
6x – 2(3x – 6) = 9
0 + 12 = 9; this can’t be true
• But, the slopes of each equation are the same:
y = 3x – 6 and 2y = 6x – 9; could have told that
Dependent equations, or Identities
• Identities are equations that are identical
• y = 3x – 6 and 2y = 6x – 9; weren’t identical, but if we had
y = 3x – 6 and 2y = 6x – 12 they would have been and we
would have gotten to a point
0=0
• In that case, we say that there are infinite solutions, or that the
system is an identity, or that we have dependent equations
Summary
• Two equations can have
– One solution (usual case)
– No solution (parallel lines)
– Infinite solutions (thy are the same line)
• When you get a solution, plug it in and verify it
• Solve by addition (elimination) or substitution
• Clear fractions (or decimals) as desired
Example
• 3x + 4y = 8
• x + 2y = 10
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Solution
• 3x + 4y = 8
• x + 2y = 10
• Solve the 2nd for x: x = 10 – 2y
• Substitute into the first: 3(10 – 2y) + 4y = 8
30 – 6y + 4y = 8
-2y = -22
y = 11; x = 10 – 2y = 10 – 22 = -12
• Check: 3(-12)+4(11) = -36 + 44 = 8; OK
(-12) + 2(11) = -12 + 22 = 10; OK
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Example
• x = 3y – 4
• x + 2y = 6
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Solution
• x = 3y – 4
• x + 2y = 6
Rewrite the first as x – 3y = -4
Subtract the 2nd from the first
-3y – 2y = -10
y = 2, subst into the 2nd, x = 2
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Example
• What if we have fractions?
x/2 + y/6 = 2/3
x/4 – y/5 = 7/4
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Solution
x/2 + y/6 = 2/3
x/4 – y/5 = 7/4
let’s clear the fractions in each. Multiply first by 6, 2nd by 20
3x + y = 4
5x – 4y = 35
Now multiply the first by 4 and add
12x + 4y = 16
5x – 4y = 35
17x = 51, x = 3, so y = -5
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Inconsistent equations?
• When you substitute, you find a contradiction
y = 3x – 6
6x – 2y = 9
Substitute the first into the second:
6x – 2(3x – 6) = 9
6x – 6x + 12 = 9 or 12 = 9
12  9
Note – the lines are parallel: Slope is 3 for both
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Consistent Equations?
• 2x = 4y + 6
• 3x – 6y = 9
• You will arrive at an identity
• Divide the second equation by 3: x – 2y = 3 or x = 2y + 3
• Substituted for x in the first:
2(2y + 3) = 4y + 6
4y + 6 = 4y + 6, an identity
• The lines are the same; all points on the line satisfy both
equations
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Example
Solve:
• y = 4x – 2
• 3x – y = 4
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Solution
• y = 4x – 2
• 3x – y = 4
Rewrite first equation as 4x – y = 2
Subtract the second equation from the first
x = 2-4 = -2
Subst into 4x – y = 2 to get y = -10
Check: -10 = -(4)(2) -2 and 3(-2) – (-10) = 4
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Inequalities
Treat Inequalities as Equations
• Solve as one would normally EXCEPT
– You cannot multiply or divide by a negative number
without changing the direction of an inequality
• If we have -2x < -4 then x > 2
• 3x < 5 + x
2x < 5 is the solution, can graph on a number line
• Open circles for < and >, close circles for  and 
Exponents
Rules
• xn is just x multiplied n times
• Product rule: xn xm = xn+m
• Power rule: (xn) m = xnm
• Quotient rule: xn / xm = xn-m
• Zero exponents: x0 = 1
• Negative exponents x-n = 1/xn and 1/x-n = xn
• In complex expressions, simplify the inside of the parentheses
first, then multiply parentheses
Polynomials
Definitions
• Contains sums of terms of the form 3xn , where 3 is the
coefficient and n is an integer  0 (no negatives or fractions)
• A monomial has one such term, binomial two, etc
• The order of a term is the sum of its exponents
• The order of a polynomial is the order of its largest term
Multiplying Polynomials
• Apply repetitive use of the distributive property; formalized in
FOIL or box foil
(3y – x)(2y + 5x) = 6y2 – 2xy + 15xy – 5x2 =
6y2 – 13xy – 5x2
Dividing by Monomials
3x4 – 4x2 – 5 = 3x3/2 – 2x – 5/x
2x
1/(2x) acts like a number multiplying each term
Scientific Notation
• 1.5 x 108
• The number multiplying must be -10< n < 10, in other words,
one place to the left of the decimal
• Use rules of exponents to find the correct exponent for the 10
• 0.000087 = 8.7 x 10-5 , move the decimal 5 places to the right
• 3456.87 = 3.45687x104, move the decimal 4 places to the left
Dividing Polynomials
• Do just like long division. Express remainder as a coefficient
(or term) divided by the denominator
Word Problems
Mixture Problems
• You have two things you need to put together to get a
required result. These can be in the form of a substance
(liquids), investments, speeds.
• They are all handled in roughly the same way.
– Identify the items
– Make a table that lists the amount of each item, the result
from that item
– Link the items in terms of two linear equations
Sample: Investment
•
Money was invested at 8% and 12% simple interest, with $3000 more invested at
8% than at 12%. If the yearly interest from both investments was $760, how much
was invested at each rate?
•
Let A be the amt at 8% and B the amount at 12%
Amt
%return
return
8% investment
A = 3000+ B
8%
0.08 A
12% investment
B
12%
0.12 B
Total
A + 3000 = B
$760
=0.08A+0.12B
Have one equation: (B + 3000) x 0.08 + B x 0.12 = 760
Answer: $2600 @ 12% and $5600 @ 8%
Constant Motion
• At 9:45 A.M., a bus left Point Reyes, California, traveling north at 40
miles per hour. Two hours later, a second bus left Point Reyes
traveling south at 35 miles per hour. At what time will the buses be
230 miles apart?
Speed
Time
Distance
Bus 1
40 mph
T
40 T
Bus 2
35 mph
T – 2 (2 hrs late) 35(T-2)
Total
230 m (add since
going opposite)
• 40T + 35 (T + 2) = 230 sum of distances is 230
• T = 4hrs, or 1:45 PM
Mixtures
• How much candy worth 32 cents a pound must be mixed with
candy costing 25 cents per pound to create 35 pounds of
mixed candy selling at 30 cents per pound?
• Let 32cent = A and 25cent = B
• A + B = 35 lbs total or A = 35 - B
• 32A+ 25B = 30(35lbs); cost of each sums to total cost
• Have two equations
Subst for A: 32 (35 –B) + 25B = 30(35) and solve
• 25 lbs @ 32 cents per pound, 10 lbs @ 25 cents per
pound
Mixtures
• A child’s piggy bank contains nickels, dimes and quarters.
There are four times as many nickels as dimes and ten fewer
quarters than nickels. The total value of all the coins is $5.30.
Find the number of each type of coin.
• Variables N, D, Q.
• N = 4D or D = N/4 and Q = N -10
• 5N + 10D + 24Q = 530; the values of each coin group add to
the total
• subst to get an equation in N (one variable)
• 6 dimes, 24 nickels, and 14 quarters.
Mixture
• How many liters of a 50% alcohol solution must be added with
80 liters of a 20% alcohol solution to make a 40% alcohol
solution?
• 50% solution amount is is A, 20% is B.
We have 80 (20%) alcohol from B and A(50%) from A
• Total is 80 + A liters
• Want (80 + A) (40%) = 80(20%) + A(50%); solve for A
• 160 liters
Geometry
• The length of a rectangular garden is 5 feet greater than the
width. The area of the rectangle is 300 square feet. Find the
length and the width.
• L=W+5
• LW = 300 = L(L + 5)
•
L = 20 ft, W = 15 ft.
Mixture
• A pool can be filled by one pipe in 3 hours and by a second
pipe in 6 hours. How long will it take using both pipes to fill the
pool?
• Rates are A and B
• 3A = full, 6B = full so, 3A = 6B or A = 2B
• Together: (A + B)t = 6B or (B + 2B) t = 6B where t is the time
to fill if both pipes work
• 2 hours together
Rates
• Two cars travel the same route, both leaving from the same
point. The slower car averages 40 miles per hour, and the
faster car averages 50 miles per hour. If the faster car leaves
2 hours after the slower car, in how many hours will the faster
car overtake the slower car?
• S slow, F fast distance traveled
• S = 40t, F = 50(t – 2), Want t for S = F
• 40t = 50(t-2)
• 8 hours
Geometry
• The length of a rectangular garden is 5 feet greater than the
width. The area of the rectangle is 300 square feet. Find the
length and the width.
Solution
• L=W+5
• LW = 300 = L(L + 5)
•
L = 20 ft, W = 15 ft.
Number
• When a number is divided by 6, the answer is 8 more than
one fifth the number. Find the number
Solution
• When a number is divided by 6, the answer is 8 more than
one fifth the number. Find the number
• n/6 = n/5 + 8
• 5n = 6n + 8
• m = -8