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University Prep Math

University Prep Math. Patricia van Donkelaar pvandonkelaar@hrsb.ns.ca https://pvandonkelaar.hrsbteachers.ednet.ns.ca. Course Outcomes By the end of the course, students should be able to Solve linear equations Solve a system of equations

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University Prep Math

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  1. University Prep Math Patricia van Donkelaar pvandonkelaar@hrsb.ns.ca https://pvandonkelaar.hrsbteachers.ednet.ns.ca • Course Outcomes • By the end of the course, students should be able to • Solve linear equations • Solve a system of equations • Identify linear, quadratic and exponential patterns • Algebraically find the equation of each type of pattern • Graph linear and quadratic functions • Transfer between the 3 forms of a quadratic • Solve quadratic equations by factoring, completing the square or by using the quadratic root formula • Solve for the vertex of a quadratic • Solve exponential equations using common bases and logs • Solve logarithmic equations • Solve simple probability problems • Use permutations and combinations to solve problems involving probability

  2. Order of Operations: When to do what B E D M A S DD I T I ON I V I S I ON R A C K E T S X P ON E N T S U B T R A C T I ON U L T I P L A C T I ON

  3. Evaluate the following expressions given the value of the variable stated. • 1) 7x–3 if x = 7 • 2) 10(x–2) if x = 4 • 3) 5r– 7t–6 if r = 2 and t = 1 • 4) 3t2 +5t–9 if t = 2 • 5) if x = 4 • 6) if j =3 Answers: 1) 46 2) 20 3) –3 4) 13 5) 26 6) 1

  4. Find the root(s) of each equation. 1) 5(x–4) = 10 2) 8w – 2 = –42 3) 4) 3x + 6 = 9x – 4 5) 6) 7m – 4 = 2m –19 7) x2 + 1 = 26 • Answers: • 6 • –5 • 3) 11 • 4) 10/6 (or 1.666…) • 5) 26 • 6) –3 • 7) 5 and –5

  5. liner not linear (quadratic) Functions A function is a relationship between two variables (where each permissible value of the independent variable corresponds to only one value of the dependent variable) y = x2 + 3 is a relationship between two variables(x and y). In this case, the function says“y is always 3 more than x times itself”. p = 1.23f is also a function. It shows the relationship between f (liters of fuel) and p (price) The simplest functions are linear

  6. Linear Functions The temperature in Dallas, Texas is 94°F. What is that temperature in degrees Celsius?

  7. Here is the graph of the (linear) relationship between Celsius and Fahrenheit. We can use it to estimate the answer…. …but if we had the mathematical formula for the relationship (EQUATION OF THE LINE) we could find the answer exactly.

  8. THE EQUATION OF A LINE The equation y = mx + b describes any straight line! For a specific line, m (slope) and b (y-intercept) are fixed or constant values, and represent actualslope value and the actualy-intercept value for that specific line. The x and y are variables that have the certain relationship as determined by the equation, so they stay in the equation.

  9. THE EQUATION OF A LINE The SLOPE (m) of a line is a number indicating its steepness. m = 0 m = −1/2 m = 1 m = 3 Each of these lines have a different slope,but the same y-intercept, b = 3.

  10. THE EQUATION OF A LINE b =5 b = 3 b = −1 b = −5 The y-INTERCEPT (b) of a line is the point where the line crosses the y (vertical) axis. Each of these lines have a different y-intercept,but the same slope m = –2.

  11. THE EQUATION OF A LINE Each distinct line has a specific slope (m) and a specific y-intercept (b). You can think of them together as the PIN of the line. Once they are know, then we have full access to all the line’s information, and can use it to solve problems. We can: • draw and use the graph of the line • find and use points on the line • write and use the equation of the line • solve problems using the linear relationship • etc…

  12. THE EQUATION OF A LINE • But how do we find thesetwo very important values? • If we have the equation, it’s easy-peasy (if the equation isy = 3x+ 5, then m = 3 and b = 5) • If we have the graph, b is usually easy-peasy (just find the y-value where the line crosses the y-axis), but mmight take some work (see next slide) • If we have at least 2 points on the line, calculate m first (see next slide), then calculate b (three slides from now) • In a word problem, the rate is m, and the initial value of the y-variable is b.

  13. THE EQUATION OF A LINE Finding slope: The slope is a measure of how muchchange there was for y (the dependentvariable) for every change in x (theindependent variable). Mathematically we divide the changein the y value between two pointsbythe change in the x value between the same two points. These two points (x1, y1) and (x2, y2) might be given, or you might find them on the graph. If you have the equation in the form y = mx + b, the m value is the slope.

  14. Here are some points we know, either from the graph or from memory: (0°C, 32°F)(−40°C, − 40°F)(10°C, 50°F) On your own… Try this with another pair of points, or use the same points in the opposite order.As long as the points you use are on this line you will ALWAYS get 9/5 as the slope! Finding the slope in our example:

  15. THE EQUATION OF A LINE Finding the y-intercept: If we have the slope m and a point on the line (x, y), sub these three values into y = mx + b and solve for b. If you have the equation in the form y = mx + b, the b value is the y-intercept. If you have the equation in a form other than y = mx + b, either put it into y = mx + b form or sub-in x = 0 and solve for y. This answer is the y-intercept b.

  16. So far we know that the slope is 9/5 and we know a point on the line (10, 50). Now we can sub-in: m = , x= 10, y= 50 On your own… Try this with another point. Remember though that m won’t change because it is a constant for this particular line. As long as the point you choose is on this line and use m = 9/5, you will ALWAYS get 32 as the y-intercept! Finding the y-intercept in our example:.

  17. USING THE EQUATION OF A LINE The temperature in Dallas, Texas is 94°F. What is that temperature in degrees Celsius? We have found m = and b = 32, so we have the following equation: or This is the equation of the line in the graph relating degrees Celsius (x or C) to degrees Fahrenheit (y or F). So, when F = 94°F, we sub this into the equation and calculate that C = 34.4°C

  18. USING THE EQUATION OF A LINE (34.4, 94) The temperature in Dallas, Texas is 94°F. What is that temperature in degrees Celsius?

  19. Cell Phone Bill – A linear function • What are two constants involved in simple cell phone billing system? • flat/base monthly fee (dollars) – b because this is the initial value • price per minute used (dollars/minute) – m because it is the rate What are two variables involved in simple cell phone billing system? • monthly usage (minutes) – x because this is the variable you can directly influence (independent variable) • monthly bill amount (dollars) – y because it depends on x (dependent variable)

  20. Cell Phone Bill – A linear function • Let’s say that it costs 20 cents per minute and that you are always charged a monthly fee of $7.00. • Questions: • Give the function thatrelates the two variables(x – number of minutes,y– monthly bill).Draw the graph of thisrelationship • If you talked for 45 minutes,what will your bill be • If your bill is $37.40, for howmany minutes were youon the phone? Answers: 1) y= 0.20x + 7 2) If x = 45minutes, y = $16 3) If y = $37.40, x = 152minutes

  21. Cell Phone Bill – A linear function • On another plan, in January you talked on your phone for 100 minutes and your bill was $30.00. In February you talked for 150 minutes, and your bill was $42.50. • Questions: • What is the charge perminute, and is the flatmonthly flat fee? • Give the function that relatesthe two variables. Draw thegraph of this relationship. • If you talked for 45 minuteson this plan, what will yourbill be? Answers: 1) m= 0.25$/minute b= 5.00$ 2) y= 0.25x + 5 3) If x = 45minutes, y = $16.25

  22. Systems of Equations But what if we want to know when these two plans cost the same amount? We will combine the two equations into a system. A system of linear equationsis a set of two simultaneous equations. The solution to a system is the point (x, y) at which both equations hold true. Graphically this is the intersection of the two lines.

  23. Systems of Equations This brace indicates theequations form a SYSTEM There are infinitely many x, y pairs which satisfy the equation3x+ 4 = y:(1, 7) or (0, 4) or (−1/3, 3) or (−100, −296) just to name a few… (this is the same as saying there are infinitely many points on the liney = 3x + 4) …but if y = −7x − 1 must ALSO be satisfied, then none of the points listed work; none satisfy BOTH equations. (the only point that satisfies both equations is the point of intersection of the two lines) So then what is the solution to ? BIG IDEA: Turn 2 equations with 2 unknowns (hard to solve)into 1 equation with 1 unknown (easy to solve)

  24. Solving Systems by Substitution At the point of intersection, both lines will have the same y value. So we can replace the y in one equation by the equivalent value of y from the other. y = 3x + 4 (– 7x – 1)= 3x + 4 –10x = 5 x = –0.5 Half way there!! Now with this half of the solution we can find the other variable. It doesn’t matter which original equation you choose: y = 3(–0.5) + 4 y = 2.5 OR y = –7(–0.5) -1 y = 2.5 Therefore, the solutionto is (−0.5, 2.5) The same! This is the ONLY (x, y) pair that satisfies BOTH equations!

  25. Solving Systems by Substitution Solution: 2) Substitute this expression into the other equation 3) Solve for the remaining variable 1) Isolate one variable in one equation.(choose wisely!) 4) Use one of the original equations to solve for the second variable Example: Solve by substitution:

  26. Solving Systems by Substitution Example cont’: Solve by substitution: The solution is (5, −2) Let’s double check our answer. The solutionx = 5 and y = −2 should satisfy both equations: Both are satisfied, so our solution is correct!

  27. Solving Systems by Elimination This is another method used to solve linear systems. It eliminates one of the variables (turns a question of 2 equations and 2 unknowns into a question with 1 variable and 1 unknown) by adding/subtracting the equations. Ex. Solve this system of equations using elimination. Let’s “mush ‘em together” (that is, let’s add the equations Um… let’s align the equations first. Let’s try to add again. Still two variables…this didn’t help!

  28. Solving Systems by Elimination Let’s add the equations. Let’s try multiplying the equations through be a number so their coefficients are opposite before addition Let’s multiply the first by 3 Let’s multiply the second by 2 Let’s add them now... Success! We eliminated y, and are left with 1 equation with 1 unknown (x), which is easy to solve! Half way there!!

  29. Solving Systems by Elimination To find y, simply plug-in x = 5 into either of the original equations: The solution to is (5, −2)

  30. Solving Systems by Elimination Solution: 4) Use one of the original equations to solve for the second variable Add to eliminate one variable 3) Solve for the remaining variable 1) Multiply and align (get two coefficients to be opposite) The solution is h = −1, f = 1 Example: Solve by elimination:

  31. Word problem example A certain Math textbook costs $10 more than 3 times the amount of an English book, before taxes. Together they total $140, before taxes. Calculate the price of each book. Solution: Let Ebe the price of the English textbook Let M be the price of the Math textbook M =10 + 3E M + E = 140 The Mathematics text costs $107.50, and the English text costs $32.50

  32. 1.Solve these systems… …using elimination: …using substitution. …using your choice: 2. Back to the cell phone example, how many minutes do you have to use for both cell phone plans to cost the same? • Answers: • a) (9, 4) c) (2, −3) e) (1, −3) • b) (−4, 7) d) (0.5, −0.5) f) (250, 700) • 2. Both plans cost the same, $15.00 when you use 40min

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