Chapter 16: Heart of Algebra

Solve this system using elimination: - A+B= 5 - 2A-3B= -5

-- A+B=5 -> 2A+2B=10 -- 2A-3B=-5 --- 2A-3B - (2A+2B) = -5 - (10) -- -5B=-15 -> B=3 ---A+B=5 -> A+3=5 -> A=2

(Comprehensive Example) In 2014, County X had 783 miles of paved roads. Starting in 2015, the county has been building 8 miles of new paved roads each year. What is the first step to solving this problem?

-Decide what variable(s) you need to define In this case, since the number of miles paved depends on the year, we can define a variable to represent the year. (N)

Many "Heart of Algebra" questions will require you to accomplish the following steps:

-Define one or more variables that represent quantities in the question -Write one or more equations, expressions, inequalities, or functions that represent the relationships described in the question. -Solve the equation -Interpret the solution in terms of what the question is asking.

Rearrange the equation "distance x rate = time." to solve for the other variables

-Distance = Time/Rate -Rate = Time/Distance

What is the first step to using algebra to analyze and solve a problem in a real life context?

-Figure out how to represent the context of the problem algebraically.

What are subsequent steps to using algebra to analyze and solve a problem in a real life context?

-Finding variables that represent quantities in context. -Writing one or more expressions, equations, and inequalities, or functions that represent the relationships described in context.

Solve this system using substitution: - A+B= 5 - 2A-3B= -5

-First --A+B=5 -> A=5-B --2A-3B=-5 -> 2(5-B)-3B=-5 --10-2B-3B=-5 -> 10-5B=-5 -- -5B=-15 -> B=3 --- A+B=5 -> A+3=5 -> A=2

(Comprehensive Example) Maizah bought a pair of pants and a briefcase at a department store. The sum of the prices of the pants, and a 9% sales tax on the briefcase. The total Maizah paid, including the sales tax, was $136.75. What was the price, in dollars, of the pants?

-First Step, Define the variables --Price of a pair of pants - P --Price of a briefcase - B -Since the sum of the prices before sales was $130.00, the equation P + B = $130 is true. -A sales tax of 9% was added to the briefcase. --Since 9% equals .09, the price of the briefcase, with tax, was B + .09B = 1.09B --There was no sales tax on the pants. --The total Maizah paid, including tax, was $136.75. --So the equation P + 1.09B = $136.75 is true. -Solve the system --P+B=130 --P+1.09B=136.75 -One way to do this is to subtract the sides of the first equation from the corresponding side of the second equation. --This gives you (P+1.09B)-(P+B)=$136.75-130 --This simplifies to .09B=6.75 --This simplifies to B=75 ---Substitute this value in the system, let's say P+B=130; and you'll find that P=55.

Solving an equation in relation to a contextual problem is only part of the problem-solving process, what is another step?

-Having to interpret the solution in the context of the question.

(Comprehensive Example) Each morning John jogs at 6 miles per hour and rides his bike at 12 miles per hour. His goal is to jog and ride his bike a total of at least 9 miles in no more than an hour. If john jogs J miles and rides his bike B miles, which of the following system of inequalities represents John's goals? A)-(J/6) + (B/12) </= 1 -J+B >/= 9 B)-(J/6) + (B/12) >/= 1 -J+B </= 9 C)-6J + 12B >/= 9 -J+B </= 1 D) -6J+12B </= 1 -J+B >/= 9

-John's goal to ride his bike B miles and Jog J miles for at least a combined 9 miles can be represented by the inequality J+B >/= 9. -Since rate x time = distance, it follows that time = distance/rate. --John jogs J miles at 6 miles per hour, so the time he jogs is equal to J miles/6 miles per hour which = J/6 hours. --Similarly, since John rides his bike B miles at 12 mph, the time he rides his bike is B/12 hours. --John's goal to complete his jog and his bike ride in no more than one hour can be represented by the inequality (J/6)+(B/12) </= 1. --- The system J+B >/= 9 & (J/6) + (B./12) </= 1 is choice A

"Heart of Algebra" questions focus on:

-Linear Equations -Systems of Linear Equations -Linear Functions

"Heart of Algebra" questions contain _____ and _____ questions and are reported on a scale of ______ (sub score).

-Multiple Choice -Gridded -1-15

What are the final steps to using algebra to analyze and solve a problem in a real life context?

-Solving the equation(s) you created in order to represent the relationships in the context. -Interpreting the Solution to the equation in terms of the context.

"Heart of Algebra" question forms:

-Straightfoward fluency exercises -Interpreting the relationship between graphical and algebraic representations -Solving as a process of reasoning

Two approaches to solving a system of algebraic equations

-Substitution --Solving for a variable of one equation and substituting the value of that variable into the other equation. -Combination --Combining the values of one equation to the values of the other.

(Comprehensive Example) A builder uses the function G defined by G(x) = 80x + 10,000 to estimate the cost G(x), in dollars, to build a 1 story home of planned floor area of x square feet. if the builder estimates that the cost to build a certain one-story home is $106,000, what is the planned floor area, in square feet, of the home?

-The estimated cost of the home, in dollars, is the output of the function G for a 1 story home of planned floor area of x square feet. -This means that the output of the function, G(x), is $106,000, and you need to find the value of the input x that gives an output of 106,000. -to do this, sub in 106,000 for G(x) in the equations that defines G: 106,000 = 80x + 10,000 -Solving for X yields x=1200 -Therefore, a one-story home with an estimated cost of $106,000 to build has a planned floor area of 1200 square feet.

(Comprehensive Example) To edit a manuscript Miguel charges $50 for the first 2 hours and $20 for each successive hour after the first 2 hours. Which of the following expresses the amount C, in dollars, Miguel charges if it takes him X hours to edit a manuscript, where X>2? A) C = 20X B) C = 20x + 10 C) C = 20x + 50 D) C = 20x + 90

-The question defines the variables C and X and asks you to express C in terms of X. Note that since the $50 that Miguel charges pays for his first two hours of editing, he charges only $20 per hour after the first two hours. Thus, if it takes X hours for Miguel to edit a manuscript he charges $20 for the reading time which is X-2 Hrs. Thus, his total charge, C in $, can be written as C = 50 + 20 (x-2), where x>2. When expanded, the equation is 50-40+20x or C = 20x + 10. Choice B)

(Comprehensive Example) -2(3x - 2.4) = -3(3x - 2.4) What is the solution to the given equation?

-The structure of the equation reveals that -2 times a quantity, 3x - 2.4, is equal to -3 times the same quantity. --This is only possible if the quantity 3x - 2.4 is equal to 0. --Thus, 3x - 2.4 = 0, or 3x = 2.4 ---Therefore, the solution is x = 0.8

(Comprehensive Example) > -2x = 4Y + 6 > 2(Y + 3) = 3X - 5 What is the solution (X,Y) to the system of equations above? A) (1,2) B) (1,-2) C) (-1,-1) D) (-1,1)

-Using substitution, since -2X = 4Y + 6, it follows that -X = 2Y + 3 --Now you can substitute -X for 2Y + 3 in the second equation. --This gives you 2(-X) = 3X - 5, which simplifies to 5X = 5, or X = 1 CONT.

(Comprehensive Example) 3([1/2]-Y) = (3/5) + 15Y What is the solution to the given equation?

-Using the distributive property to expand the left-hand side of the equation gives (3/2) - 3Y = (3/5) - 15Y -Adding 3Y to both sides of the equation and then subtracting (3/5) from both sides of the equation gives (3/2) - (3/5) = 18Y. --The equation may be easier to solve if it's transformed into an equation without fractions; to do this, multiply each side of (3/2) - (3/5) = 18Y by 10, which is the least common multiple of the denominators 2 and 5. --This gives (30/2) - (30/5) = 180Y, which can be simplified further to 15-6 = 180Y, or 9 = 180Y. ---Therefore, Y = (1/20).

Remember that solving systems of equations involves many of the same steps to solve other linear equations and inequalities like in that you must what?

-define variables -create equations to represent relationships -Solve those relationships -solve those equations -interpret the solution

Systems of Linear Equations and Inequalities in Context

This section asks you to define more than one variable and create more than one equation or inequality to represent a context and answer a question.

Part of what the SAT assesses is your ability to decide when using a what?

calculator

Some questions will have answers that contain two parts, how can you use this to your advantage?

By working out one part of the answer and eliminating answers that don't work with what you solve.

(Comprehensive Example) In 2014, County X had 783 miles of paved roads. Starting in 2015, the county has been building 8 miles of new paved roads each year. Which of the following functions (F) gives the number of miles of paved road there will be in County X? A) F(N) = 8 + 783N B) F(N) = 2014 + 783N C) F(N) = 783 + 8N D) F(N) = 2014 + 8N

C) F(N) = 783 + 8N

(Comprehensive Example) In 2014, County X had 783 miles of paved roads. Starting in 2015, the county has been building 8 miles of new paved roads each year. At this rate, in which year will county X first have at least 1000 miles of paved roads?

First step, create and solve an inequality. N = # of years post 2014 783 + 8N = F(N) = # of miles of paved road N years past 2014. The inequality should look like 783 + 8N >/= 1000 Solving for the inequality should yield an answer of N = 27.125 Since the context implies that it is looking for the nearest WHOLE YEAR that county X will first have AT LEAST 1000 miles of new road, therefore, it is appropriate to adjust the answer to N = 28

(Comprehensive Example) In 2014, County X had 783 miles of paved roads. Starting in 2015, the county has been building 8 miles of new paved roads each year. At this rate, how many miles of paved road will county X have in 2030?

If County X already had 783 miles of paved road and paves new road at a rate of 8 miles per year (post 2014), then defining the variables of the context should lead you to an equation of F(N) = 783 + 8N. N being the number of years after 2014. 2030-2014 = 16 = N. That means County X will have a total of F(N) = 783 + 8(16) = 911 miles of new paved roads in 2030.

While a calculator is permitted on one portion of the SAT math test, what is important to remember?

That some questions can be solved more efficiently without using a calculator and your ability to choose when and when not to use one is part of what the SAT assesses.