Transcript Algebra Released Questions 19-41

Standard 6.0
19. What is the y-intercept of the graph of
4x + 2y = 12
a)
b)
c)
d)
-4
-2
6
12
Answer
• 1st, place the standard form equation in
point slope intercept form (y = mx + b). By
doing this, “b” will be your answer to the
question.
4x + 2y = 12
4x – 4x +2y = 12 – 4x
2y/ = -4x/ + 12/
2
2
2
y = -2x + 6
• -2 = m or the slope, and 6 = b or the yintercept. The y-intercept is your answer.
• ∴ the answer is “c”
Standard 6.0
20.Which inequality is shown on the graph
below?
a) y < ½ x – 1
b) y ≤ ½ x – 1
c) y > ½ x – 1
d) y ≥ ½ x – 1
• 1st, eliminate two answers with what you
know. If the slope is a solid line then the
inequality must be ≥ or ≤ . Regular >
and/or < are represented by dotted lines.
• You are left with “b” and “d”. Look at
which side of the line is shaded. If the area
is shaded below the line, then y is less than
or equal to (y≤ ), if shaded above the line
then y is greater than or equal to (y≥). In
this case, it is shaded above.
• ∴ y ≥ ½ x – 1, so your answer is “d”
Standard 6.0
21. Which best represents the graph y = 2x – 2?
• 1st, eliminate what you know. The equation
is in point slope form (y= mx + b). 2 = m or
your slope. Since it is positive, the line goes
from low left to high right. This eliminates
“b” and “c”
• Next, - 2 = b or your y intercept (where the
line crosses the y-axis. In this case, “a” has
the slope crossing the y-axis at (0,-2) while
“d” has it crossing at (0,2). Knowing that
you are looking for is – 2, you have your
answer.
• ∴ “a” is your answer.
Standard 7.0
22.Which point lies on the line defined by
3x + 6y = 2.
a)
b)
c)
d)
(0, 2)
(0, 6)
(1, - 1/6 )
(1, - 1/3 )
•The best and easiest way to solve this
problem is to insert the numbers for x and
y. Remember, the numbers given are (x, y).
a)
3(0) + 6(2) = 2
12 ≠ 2
b)
3(0) + 6(6) = 2
36 ≠ 2
c)
3(1) + 6(- 1/6) = 2
3–1=2
d)
3(1) + 6(- 1/3) = 2
3–2≠2
• ∴ “c” is your answer.
Standard 7.0
23. What is the equation of the line that has
a slope of 4 and passes through the point
(3, – 10)?
a)
b)
c)
d)
y = 4x – 22
y = 4x + 22
y = 4x – 43
y = 4x + 43
• 1st, don’t get distracted by the slope of 4
comment for you already know (4x)
• Simply plug (x , y) or (3, -10) into equations.
a)
b)
-10 = 4(3) – 22
-10 = 4(3) + 22
-10 = 12 – 22
-10 = 12 + 22
-10 = -10
-10 ≠ 32
c)
-10 = 4(3) – 43
-10 = 12 – 43
-10 ≠ -21
• ∴ the answer is “a”
d)
-10 = 4(3) + 43
-10 = 12 + 43
-10 ≠ 55
Standard 7.0
24. The data in the table show the cost of renting a
bicycle by the hour, including a deposit.
Hours (h)
2
5
8
Cost in Dollars (c)
15
30
45
• If hours, h, were graphed on the horizontal axis
and cost, c, were graphed on the vertical axis, what
would be the equation of a line that fits the data?
a) c = 5h
c) c = 5h + 5
b) c = 1/5h + 5
d) c = 5h – 5
• 1st remember that horizontal axis is the xaxis and the vertical axis is the y-axis.
• Looking at the chart, pick one pair and plug
into the equation.
b)
a)
1/ (2) + 5
15
=
5
15 = 5(2)
2/
15
≠
5
5
15 ≠ 10
c)
d)
15 = 5(2) + 5
15 = 10 + 5
15 = 15
• ∴ the answer is “c”.
15 = 5(2) – 5
15 = 10 - 5
15 ≠ 5
Standard 8.0
25. The equation of line l is 6x + 5y = 3, and
the equation of line q is 5x – 6y = 0.
Which statement about the two lines is
true?
a)
b)
c)
d)
Lines l and q have the same y - intercept
Lines l and q are parallel
Lines l and q have the same x intercept
Lines l and q are perpendicular.
• 1st, place both equations in slope
intercept form.
Looking at the
• slope, to determine
6x + 5y = 3
6x – 6x + 5y = 3 – 6x
1/ • 5y = -6/ x + 3/
5
5
5
y = -6/5 x + 3/5
5x – 6y = 0
5x – 5x – 6y = 0 -5x
-1/ • - 6y = -5x • - 1/
6
6
y = 5/6 x
if a line is parallel
or perpendicular
you look at the
slope (m) in the
equation. In this
case, the slopes
are an inverse with
an opposite sign.
If this is the case,
the lines are
perpendicular.
Standard 8.0
26. Which equation represents a line that is
parallel to y = - 5/4 x + 2?
a)
b)
c)
d)
y = - 5/4 x + 1
y = - 4/5 x + 2
y = 4/5 x + 3
y = 5/4 x + 4
• Do not waste time on this problem. The
problem is already in point slope form
( y = m x + b ) and the “m” or slope is
what you need.
• The problem asked for parallel lines .
When comparing lines and their
equation, the quickest way to determine
if they are parallel is if they have the
same slope or “m” in the above equation.
• ∴ the answer is “a”.
Standard 9.0
27. Which graph best represents the
solution to this system of inequalities.
{
2x≥ y – 1
2x – 5y≤10
• You want to place it into point slope
intercept form (y = mx + b).
• Remember, if you have a negative “y” at
the end, you will need to switch the
direction of the sign.
•
2x ≥ y -1
2x -5y ≤ 10
• 2x +1 ≥ y
-5y ≤ -2x +10
•
y≤ 2x + 1
y ≥ 2/5 x – 2
• Use the y-intercept (b) to determine the
line. First slope is less than 1 or colored in
below the line. The second is greater than
or above. The common highlighted area
is your answer. ∴ the answer is “c”
Standard 9.0
28.What is the solution to this system of
equations?
y = - 3x – 2
6x + 2y = -4
{
a) ( 6, 2 )
b) ( 1, -5 )
c) no solution
d) infinitely many solutions
• First, eliminate what you can. Look at
the first answer (6,2), if you plug the 6
into y = - 3x – 2 it would be impossible to
solve, so “a” is ruled out.
• Try the second answer. If you notice it
works for both equations, so you can
eliminate “c” or no solutions.
• Now look at the equations. If you
noticed, they form the same line. If they
are the same line, then they have infinite
number of solutions. ∴ although “b” is
one solution the answer is “d” for any
point on the slope is a solution.
Standard 9.0
29.Which ordered pair is the solution to the
system of equations below?
x + 3y = 7
x + 2y = 10
{
a)
b)
c)
d)
( 7/2 , 13/4 )
( 7/2 , 17/5 )
( -2 , 3 )
( 16 , -3 )
• When finding the solution, remember
that it must work for both equations.
• Suggestion, since fractions take longer
when solving, try the whole numbers
first.
• For this problem, “d” is your answer.
Replace x and y with the ordered pair
(16, -3).
•
x + 3y = 7
x + 2y = 10
• 16 + 3 (-3) = 7
16 + 2 (-3) = 10
•
16 – 9 = 7
16 – 6 = 10
•
7=7
10 = 10
Standard 9.0
30. Marcy has a total of 100 dimes and
quarters. If the total value of the coins is
$14.05, how many quarters does she
have?
a)
b)
c)
d)
27
40
56
73
• 1st find out how many quarters go into
$14.05. Divide 14.05 by .25 which equals
56.2. ∴, “c” and “d” are eliminated for
being too high. Remember you have a
total of 100 coins.
• Down to two, use logic or just multiply
.25 times 27 and times 40. Simply, you
need to finish with a .05. This should tell
you that multiples of 10 will not work.
.25 • 40 = 10 or $10. You have no nickel
or .05 thus, it eliminates choice “b”
• ∴ “a” is your only logical choice.
Standard 10
31.
a)
5x3
10x7
2x4
b)
1
2x
c)
1
5x
4
d)
4
4
x
5
• 1st remember to look at like terms and
treat them like separate problems. You
can do this since both the numerator and
denominator are being multiplied.
• Reduce
5 1

10 2
x
3
x
7

1
x
4
• After reducing your like terms, combine
your answers by multiplication to solve.
1 1
1


2 x4 2 x4
• ∴ your answer is “b”.
Standard 10.0
32. (4x2 – 2x + 8) – (x2 + 3x – 2)
a)
b)
c)
d)
3x2 + x + 6
3x2 + x + 10
3x2 – 5x + 6
3x2 – 5x + 10
• 1st, always remember that a subtraction
sign or negative sign in front of ( ) means
all signs change within the ( ) that follows
• After changing the signs, you want to set it
up like an addition problem. Line up
your like terms and add. CAUTION,
make sure you do the 1st step correctly!!!
•
4x2 – 2x + 8
•
(+) – x2 – 3x + 2
•
3x2 – 5x + 10
• ∴ your answer is “d”
Standard 10.0
33. The sum of two binomials is 5x2 – 6x. If
one of the binomials is 3x2 – 2x, what is
the other binomial?
a)
b)
c)
d)
2x2 – 4x
2x2 – 8x
8x2 + 4x
8x2 – 8x
• 1st, remember the word “sum” is the
answer to an addition problem.
• In this type of problem, you need to
subtract to find the missing expression.
• The expression after “is” (or equal) will
be on top. WATCH OUT OF THE
SIGNS!!!!!
•
5x2 – 6x
5x2 – 6x
• (-) 3x2 – 2x
(+) - 3x2 + 2x
•
2x2 – 4x
2x2 – 4x
• ∴ your answer is “a”
Standard 10.0
34. Which of the following expressions is
equal to
(x + 2) + (x – 2)(2x + 1)
a)
b)
c)
d)
2x2 – 2x
2x2 – 4x
2x2 + x
4x2 + 2x
• Remember your order of operations.
1st, use the distribute property to find
the product of (x + 3)(2x + 1).
• (x – 2)(2x + 1)
2x2 + x – 4x – 2
•
2x2 – 3x – 2
• Now take (2x2 – 3x – 2) and add it to
(x + 2).
2x2 – 3x – 2
(+) x + 2
2x2 – 2x + 0
• Drop the zero and you have your
answer. ∴ your answer is “a”
Standard 11.0
35. Which is the factored form of
3a2 – 24ab + 48b2
a)
b)
c)
d)
(3a – 8b)(a – 6b)
(3a – 16b)(a – 3b)
3(a – 4b)(a – 4b)
3(a – 8b)(a – 8b)
• 1st, factor the coefficients to make easier.
• 3a2 – 24ab + 48b2 has a 3 in common for
each so take it out. 3(a2 – 8ab + 16b2).
Now factor. Make sure to look at the
second sign, if it is a plus then both signs
in the ( ) are minus since the middle term
has a minus 8ab. Since they are both
negative numbers, what number times
another can also be added together to
make -8 (the answer -4)
• 3(a2 – 8ab + 16b2)
• 3(a – 4b)( a – 4b)
• ∴the answer is “c”
Standard 11.0
36. Which is a factor of x2 – 11x + 24?
a)
b)
c)
d)
x+3
x–3
x+4
x–4
• When you see this problem, they are
asking for you to factor the expression.
After factoring, they want you to look at
just one of the two ( ) to determine the
answer.
• x2 – 11x + 24
• (x
) (x
)
• (x – ) (x – )
• (x – 8) ( x – 3 )
• ∴ looking at the possible answers, “b”
would be your choice since (x – 3) is a
factor of the expression.
Standard 11.0
37. Which of the following shows
9t2 + 12t + 4 factored completely.
a)
b)
c)
d)
(3t + 2)2
(3t + 4)(3t + 1)
(9t + 4)(t + 1)
9t2 + 12t + 4
• Like the previous question, you are
factoring out the expression. You might
notice the first and last are squares
(9t2 = 3t • 3t) and (4 = 2 • 2). Also, notice
that you have a plus (+) in front of the 4
which means your signs are the same.
• 9t2 + 12t + 4
• ( 3t + 2 )( 3t + 2 )
• If you are unsure of the answer, use FOIL.
• If you notice (3t + 2) is repeated which
means it can be rewritten as (3t + 2)2.
• ∴ the answer is “a”
Standard 14.0
38. If x2 is added to x, the sum is 42.
Which of the following could be the
value of x?
a)
b)
c)
d)
-7
-6
14
42
•
Solving quadratic equation you 1st set up the
problem, x2 + x = 42. There is two ways to
solve this problem; trial and error (input
your answers until you find one that works)
or factoring.
•
•
•
•
•
•
x2 + x = 42
(- 7)2 + (-7) = 42
49 - 7 = 42
42 = 42
x2 + x = 42
x2 + x – 42 = 0
(x + 7) (x – 6) = 0
x+7=0 x–6=0
x = -7
x=6
- 7 is a possible answer ∴ “a” is your answer.
Standard 14.0
39. What quantity should be added to both
sides of this equation to complete the
square?
x2 – 8x = 5
a)
b)
c)
d)
4
–4
16
– 16
• The key phrase is “completing the square”
The quadratic equation (ax2 + bx + c = 0)
is in the format (ax2 + bx = - c) needed,
x2 – 8x = 5, to complete the square.
• The next step is to make sure the “a” is 1,
if it was not then you would have to
multiply 1/a to each side.
• Then, using “b” (-8) add (b/2)2 or (-8/2)2 to
each side. Divide – 8 by 2 then square the
answer -4. At this point you can stop.
After (-4)(-4) = 16 you get the answer to
the question, what quantity should be
added. ∴ the answer is “c” or 16.
Standard 14.0
40. What are the solutions for the
quadratic equation x2 + 6x = 16?
a) - 2, - 8
b) - 2, 8
c) 2, -8
d) 2, 8
• Two of the three methods to find your
answer in quadratic equations are below.
The answer is “c”.
2
x  6 x  16
• Factoring method:
2
x
 6 x  16  0
• x+8=0
x–2=0
•
x = -8
x = 2 ( x  8)( x  2)
• Quadratic Formula: x 2  6 x  16  0
 6  10  6  10
 6  62  ((4)(1)( 16))
2
(2)(1)
2
 16
2
• x= -8
4
2
x=2
6  (36  64)
2
 6  (10)
2
Standard 14.0
41. Leanne correctly solved the equation
x2 + 4x = 6 by completing the square.
Which equation is part of her solution?
a)
b)
c)
d)
(x + 2)2 = 8
(x + 2)2 = 10
(x + 4)2 = 10
(x + 4)2 = 22
•
Complete the square (until you find the
equation that is part of the solution).
•
x2 + 4x = 6
• x2 + 4x + (4/2)2 = 6 + (4/2)2
• x2 + 4x + (2)2 = 6 + (2)2
•
x2 + 4x + 4 = 6 + 4
•
(x + 2)(x+2) = 10
•
(x + 2)2 = 10
• At this point you can stop for since the
question is not asking for a final answer.
∴ the answer is “b”.