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Math Refresher III: Systems of Equations, Inequalities

RPAD Welcome Week August 17-23, 2013. Math Refresher III: Systems of Equations, Inequalities. Gang Chen Assistant Professor gchen3@albany.edu. Gang Chen Public Finance and Budgeting MPA course: PAD 501 (Financial management) PAD 642 (Public Budgeting). Path of math learning.

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Math Refresher III: Systems of Equations, Inequalities

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  1. RPAD Welcome Week August 17-23, 2013 Math Refresher III: Systems of Equations, Inequalities Gang Chen Assistant Professor gchen3@albany.edu

  2. Gang Chen • Public Finance and Budgeting • MPA course: • PAD 501 (Financial management) • PAD 642 (Public Budgeting)

  3. 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

  4. Functions Takes a number and adds 3 Terminology 8, 10, 11, 12 5, 7, 8, 9 X Domain Input Independent variable f (or g, or h as name of function) Rule X + 3 Y Range Output Dependent 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.)

  5. Functions Examples • Given this function, find each of the following:

  6. Functions Exercises • Given this function, find each of the following:

  7. Solving Systems of Equations 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. Example • Solve (i.e., find solutions): • We can find infinitely many solutions to this equation.

  8. Solving Systems of Equations A system of equations means that there is more than one equation related to one another. Example • Solve: • The solution to this system is . • Prove it by plugging these values in the system. 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.)

  9. Solving Systems of Equations Five steps of elimination by addition: Write the equations in standard form like . Multiply (if necessary) the equations by constants so that the coefficients of the or the variable are the negatives of one another. Add the equations from step 1. Solve the equations from step 2. Substitute the answer from step 3 back into one of the original equations, and solve for the second variable. Example • Solve: • Write them in standard form. • Multiply (the second equation by -2 so that the y-coefficients are the negatives of one another.) • Add. • Solve. 5. Substitute back into an original equation. . Solution to system is

  10. Solving Systems of Equations Example (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. (0, 6) (2, 3) (, 0) (4, 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.

  11. Solving Systems of Equations Five steps of elimination by addition: Write the equations in standard form like . Multiply (if necessary) the equations by constants so that the coefficients of the or the variable are the negatives of one another. Add the equations from step 1. Solve the equations from step 2. Substitute the answer from step 3 back into one of the original equations, and solve for the second variable. Exercise • Solve: • Write them in standard form. • Multiply (the second equation by so that the y-coefficients are the negatives of one another.) • Add. • Solve. 5. Substitute back into an original equation. . Solution to system is

  12. Solving Systems of Equations (0, 7) Exercise (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. (-2, 1) (, 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.

  13. Solving Systems of Equations Three steps of elimination by substitution: Find any variable with a coefficient of 1, or make any variable so, and isolate it in one equation like . Use the equation having the variable with a coefficient of 1 to replace that variable in the other equation. Finish the problem as before by substituting back into an original equation. Example • Solve: • Find (or make) any variable having a coefficient of 1, and isolate it. • Use the isolated variable with a coefficient of 1 to replace that in the other equation. • Finish the problem. . Solution to system is

  14. Solving Systems of Equations Three steps of elimination by substitution: Find any variable with a coefficient of 1, or make any variable so, and isolate it in one equation like . Use the equation having the variable with a coefficient of 1 to replace that variable in the other equation. Finish the problem as before by substituting back into an original equation. Exercise • Solve: • Find (or make) any variable having a coefficient of 1, and isolate it. • Use the isolated variable with a coefficient of 1 to replace that in the other equation. • Finish the problem. . Solution to system is

  15. Solving Inequalities—First Degree Examples 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.” Both expressions and have the same meaning. But is a better way because it clearly visualizes the direction of difference like the number line. Number line -2 < 3

  16. Solving Inequalities—First Degree 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. Examples • Solve: then and • Solve: then and . 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.

  17. Solving Inequalities—First Degree Rule 2: Reverse the direction of an inequality symbol whenever an inequality is multiplied or divided by the same negative number. Examples then and • , multiplied by 4 then and • -5 then and (Or • , multiplied by -2 then and

  18. Solving Inequalities—First Degree 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. Example • Solve: • Graphically represent the solution

  19. Solving Inequalities—First Degree 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. Exercise • Solve: • Graphically represent the solution

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