Algebra 1, 2 – systems of equations

Q: Red pens cost twice as much as blue pens. Two blue pens and three red pens cost a total of $2.32. How much does a blue pen cost?

 Answer:  $0.29

 Explanation:  Let r be the cost of a red pen, and let b be the cost of a blue pen. 

Then, r = 2b and 2b + 3r = 2.32. Solve this system of equations. 

By substitution,

r + 3r = 2.32

4r = 2.32

r = 0.58

Plug r= 0.58 into the first equation (or second, but I like the first better):

0.58 = 2b

0.29 = b

 You can also use other algebraic methods such as elimination, Cramer’s Rule, etc.  Or use my TI-84 Plus program CRAMER once you rearrange the system to say: 2b – r = 0 and 2b + 3r = 2.32.

 

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SAT, Algebra 1 – percent change

Q: Last year, Emmy was 40 inches tall.  This year, she is 42 inches tall.  By what percent has her height increased?

 Answer: 5%

 Explanation:  Percent change is calculated by the following formula:

 (New – Original)/Original  x 100%

 (42 – 40)/40 = 2/40 = .05

 .05 x 100% = 5%

 Since it is positive, this is a percent increase.

My TI-84 Plus program PRCNTCHG will calculate this, too.

 

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Algebra 2 – linear functions

Q:  f(x) is a linear function with a y-intercept of 6. If f(-2) = 9, what is f(2)?

Answer:   3

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Explanation:

f(x) = mx + b              (because it is linear)

m is the slope of the line passing through (0,6) and (-2,9).

m = (9-6) / (-2-0)

m = – 3/2

b is the y-intercept.  b = 6.

So, f(x) = – 3/2 x + 6

Now, find f(2):

f(2) = (- 3/2)(2) + 6 = 3

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Algebra 1, 2 – solving quadratic equations

Q: The square of a positive number is equal to 5 more than the product of 4 and the same number. What is the number?

Answer: 5

Explanation:

2014-09-04_231259

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Algebra 1 – Find the equation of a line

Q:  Find the equation of the line that has the following intercepts:

x-intercept: 5,   y-intercept: -3

Answer:  y = 3/5 x – 3

Explanation:  The x-intercept is the point (5, 0).  The y-intercept is the point (0, -3).  Calculate the slope of the line passing through these points:

m= (0- -3)/(5 – 0) = 3/5

Using slope intercept form (y = mx + b), the equation is y = 3/5 x + b.  We just need to find b.  Or do we?  We could plug in an ordered pair and solve for b, but it’s not necessary.  We already have b, because in slope-intercept form, b always stands for the y-intercept.  So, b = -3.

The equation is: y = 3/5 x – 3

My TI-84 Plus program ‘LINES’ will find this equation for you, given the two intercepts.

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SAT / Algebra 1: Ratios

Q:  In my box of cereal, the ratio of flakes to raisins is 9:2. If there are a total of 220 flakes and raisins in the box, how many of these are flakes?
 
Answer:  180
Explanation:  Solve the equation: 9x + 2x = 220 to get x = 20.  ‘Flakes’ are the 9x part of the equation.  9(20) = 180.  Or, use my TI-84 Plus program ‘RATIO’.
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Algebra 1, 2 – functions

Q: Which of the following represents a function?  (Choose all correct answers.)
 
a. {(4,2), (6,2), (-1,5), (3,-2), (-4,-7)}
b. {(4,2), (5,1), (-1,5), (5,-2), (4,-2)}
c. {(-1,-1), (1,1), (4,4), (3,3), (6,6)}
d. {(1,2), (3,2), (5,2), (7,2), (9,2)}
 
Answer: a, c, and d
Explanation:  A function must have only one y-value for each x-value.  Choice b has two points with the same x:  (4,2) and (4, -2) (and also two others).  All of the other choices have no repeated values for x, so they are functions.
 
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SAT – systems of equations

Q: A phone call costs c cents for the first 10 minutes and n cents per minute for additional minutes. A 20-minute phone call costs 50 cents, and a 40-minute phone call costs 90 cents. What is the value of c?
 
Answer: 30
Explanation:  A 20-minute call costs c cents for the first 10 minutes and n cents per minute for each of the next 10 minutes.  So, c+10n=50.  A 40-minute call costs c cents for the first 10 minutes and n cents per minute for each of the next 30 minutes: c+30n=90.  Solve this system of equations.  Use my TI-84 Plus program CRAMER or an algebraic method such as substitution, elimination, Cramer’s Rule, etc.  c=30.
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Algebra 1, 2 – systems of equations

Q: 100 people are to be seated at 22 tables. Some tables seat 4 people and some seat 6 people. If no seats are empty, how many tables seating 4 people are there?
 
Answer: 16
Explanation:  Solve the system: x + y = 22; 4x + 6y = 100.  My ‘CRAMER’ program will solve this for you! The number of 4-person tables is the x-value of the solution.
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Algebra 1 – distance

Q: To the nearest tenth, what is the distance between the points (-6,2) and (6,5)?
 
Answer:  12.4 
(Use the distance formula, and round to the nearest tenth.)
My TI-84 Plus program DISTANCE will find this value.
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