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RPAD Welcome Week August 17-23, 2013. Math Refresher III: Systems of Equations, Inequalities. Gang Chen Assistant Professor [email protected]. Gang Chen Public Finance and Budgeting MPA course: PAD 501 (Financial management) PAD 642 (Public Budgeting). Path of math learning. - PowerPoint PPT Presentation
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Math Refresher III:
Systems of Equations, Inequalities
Gang ChenAssistant Professor
RPAD Welcome WeekAugust 17-23, 2013
• Gang Chen• Public Finance and Budgeting• MPA course:– PAD 501 (Financial management)– PAD 642 (Public Budgeting)
Path of math learning
• Remember the basic rules• Follow step-by-step procedures• Practices• Look for resources
• Suggested book: Barron’s Forgotten Algebra• Today’s topics cover Chapter 25, 26, 28
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FunctionsTerminology
5, 7, 8, 9 8, 10, 11, 12Takes a number
and adds 3
XDomainInputIndependent variable
f (or g, or h as name of function)
RuleX + 3
YRangeOutputDependent variable
A function is a rule that assigns to each element in the domain one and only one element in the range. (Unless specified, the domain of a function is the set of all real numbers.)
𝑦= 𝑓 (𝑥 )=𝑥+3 h𝑤𝑖𝑡 𝐷={5 ,7 ,8 ,9 } 3, h h 5, 7, 8, 9.𝑦 𝑒𝑞𝑢𝑎𝑙𝑠 𝑓 𝑜𝑓 𝑥 𝑒𝑞𝑢𝑎𝑙𝑠 𝑥 𝑝𝑙𝑢𝑠 𝑤𝑖𝑡 𝑑𝑜𝑚𝑎𝑖𝑛 𝐷 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑡 𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑎𝑛𝑑
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FunctionsExamples
• Given this function, find each of the following:𝑦= 𝑓 (𝑥 )=𝑥+3
𝑓 (7)
𝑓 (9)
𝑓 −1(11)
¿7+3=10
¿9+3=12
11=𝑥+3 11𝑓 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 h 11, h ?𝑖𝑓 𝑡 𝑒 𝑦 𝑣𝑎𝑙𝑢𝑒 𝑖𝑠 𝑤 𝑎𝑡 𝑖𝑠 𝑥
𝑥=11−3=8
7𝑓 𝑜𝑓
9𝑓 𝑜𝑓
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FunctionsExercises
• Given this function, find each of the following:
𝑦=𝑔 (𝑥 )=1+2𝑥2
𝑔 (3)
𝑔 (−2)
𝑔−1(19)
¿1+2∗32=1+2∗9=19
¿1+2∗ (−2 )2=1+2∗4=9
19=1+2∗𝑥2 𝑥2=9
√𝑥2=√9 𝑥=±3 h 19, h ?𝑖𝑓 𝑡 𝑒 𝑦 𝑣𝑎𝑙𝑢𝑒 𝑖𝑠 𝑤 𝑎𝑡 𝑖𝑠 𝑥
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Solving Systems of Equations
Example
• Solve (i.e., find solutions):
o We can find infinitely many solutions to this equation.
An equation that has the form with , , and being real numbers, and not both zero, is a linear equation in two variables.The solutions to a system of equations are the pairs of values of and that satisfy all the equations in the system.
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Solving Systems of Equations
Example
• Solve:
o The solution to this system is .
o Prove it by plugging these values in the system.
A system of equations means that there is more than one equation related to one another.
How can we find this solution?There are numerous ways to do it. But we will cover only two methods—Elimination by addition (or by substitution.)
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Solving Systems of EquationsExample
• Solve:
1. Write them in standard form.
2. Multiply (the second equation by -2 so that the y-coefficients are the negatives of one another.)
3. Add.
4. Solve.
Five steps of elimination by addition:1. Write the equations in standard
form like .2. Multiply (if necessary) the
equations by constants so that the coefficients of the or the variable are the negatives of one another.
3. Add the equations from step 1.4. Solve the equations from step 2.5. Substitute the answer from step
3 back into one of the original equations, and solve for the second variable.
5. Substitute back into an original equation.
.
Solution to system is
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Solving Systems of EquationsExample (continued)
• System:
• From an algebraic point of
view, is the solution to this
system.
• From a geometric point of
view, (2, 3) is the point of
intersection for two lines
whose equations are given
above.
3 𝑥+2 𝑦=12 𝑦=2 𝑥−1
(2, 3)
How to draw each line easily?Find x-intercept and y-intercept by plugging zero in x or y. And connect those intercepts.
(0, 6)
(4, 0)(, 0)
(0, -1)
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Solving Systems of EquationsExercise
• Solve:
1. Write them in standard form.
2. Multiply (the second equation by so that the y-coefficients are the negatives of one another.)
3. Add.
4. Solve.
Five steps of elimination by addition:1. Write the equations in standard
form like .2. Multiply (if necessary) the
equations by constants so that the coefficients of the or the variable are the negatives of one another.
3. Add the equations from step 1.4. Solve the equations from step 2.5. Substitute the answer from step
3 back into one of the original equations, and solve for the second variable.
5. Substitute back into an original equation.
.
Solution to system is
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Solving Systems of EquationsExercise (continued)
• System:
• Given the algebraic solution to
system (, show and verify that
the solution point (-2, 1) also
makes sense from a geometric
point of view.
5 𝑦+5=−5 𝑥 3 𝑥− 𝑦=−7
(-2, 1)
(0, 7)
(, 0) (-1, 0)
(0, -1)
How to draw each line easily?Find x-intercept and y-intercept by plugging zero in x or y. And connect those intercepts.
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Solving Systems of EquationsExample
• Solve:1. Find (or make) any variable
having a coefficient of 1, and isolate it.
2. Use the isolated variable with a coefficient of 1 to replace that in the other equation.
3. Finish the problem.
Three steps of elimination by substitution:1. Find any variable with a coefficient of 1,
or make any variable so, and isolate it in one equation like .
2. Use the equation having the variable with a coefficient of 1 to replace that variable in the other equation.
3. Finish the problem as before by substituting back into an original equation.
3 𝑥+2(2 𝑥−8)=12
.
Solution to system is
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Solving Systems of EquationsExercise
• Solve:1. Find (or make) any variable
having a coefficient of 1, and isolate it.
2. Use the isolated variable with a coefficient of 1 to replace that in the other equation.
3. Finish the problem.
Three steps of elimination by substitution:1. Find any variable with a coefficient of 1,
or make any variable so, and isolate it in one equation like .
2. Use the equation having the variable with a coefficient of 1 to replace that variable in the other equation.
3. Finish the problem as before by substituting back into an original equation.
3 𝑥+4 (3𝑥+1 )=−26
.
Solution to system is
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Solving Inequalities—First DegreeExamples for inequality signs
• 2 < 3 is read “2 is less than 3.”
• 5 > 1 is read “5 is greater than 1.”
• is read “ is less than or equal to 4.”
• is read “ is greater than or equal to 7.”
-2 3<
Number line
Both expressions and have the same meaning.But is a better way because it clearly visualizes the direction of difference like the number line.
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Solving Inequalities—First DegreeExamples
• Solve:
then
and
• Solve:
then
and .
To solve a first-degree inequality, find the values of that satisfy the inequality. The basic strategy is the same as that used to solve first-degree equations.
Rule 1: A term may be transposed from one side of the inequality to the other by changing its sign as it crosses the inequality sign.
• Graphically represent the solutions
The heavy line indicates that all numbers to the left of 2 (or to the right of 3) are part of the answer.The open circle indicates that 2 is not part of the answer. The closed circle indicates that 3 is a part of the answer.
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Solving Inequalities—First DegreeExamples
then and • , multiplied by 4then and
Rule 2: Reverse the direction of an inequality symbol whenever an inequality is multiplied or divided by the same negative number.
• -5then and (Or • , multiplied by -2thenand
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Solving Inequalities—First DegreeExample
• Solve:
Rule 1: A term may be transposed from one side of the inequality to the other by changing its sign as it crosses the inequality sign.Rule 2: Reverse the direction of an inequality symbol whenever an inequality is multiplied or divided by the same negative number.
• Graphically represent the solution
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Solving Inequalities—First DegreeExercise
• Solve:
Rule 1: A term may be transposed from one side of the inequality to the other by changing its sign as it crosses the inequality sign.Rule 2: Reverse the direction of an inequality symbol whenever an inequality is multiplied or divided by the same negative number.
• Graphically represent the solution
