Download Study Island Algebra 1 Critical Topics - Milan C

Study Island Algebra 1 Critical Topics
Simplify Expressions
1. Simplify.
A. 10x2 - 26xy - 12y2
B. 10x2 + 11xy - 6y2
C. 10x2 - 15xy - 6y2
D. 10x2 + 26xy + 12y2
Patterns
2. What is the explicit formula for this sequence?
45 , 48 , 51 , 54 , 57 , ...
A. an = 48 - 3n
B. an = 42 + 3n
C. an = 47 - 2n
D. an = 43 + 2n
Patterns
3. What is the explicit formula for this sequence?
-4 , 4 , 12 , 20 , 28 , ...
A. an = -12 + 8n
B. an = -11 + 7n
C. an = 4 - 8n
D. an = 3 - 7n
Quadratic Equations
4. Solve for x in the following equation.
A.
B.
C.
D.
Linear Equations
5.
(1/5) (2x + 7) = 4
What is the first step when solving the above equation for x?
A. Multiply each side by 5.
B. Subtract 2x from each side.
C. Divide each side by 5.
D. Subtract 5 from each side.
Patterns
6. Define the sequence below recursively.
an = 5n - 2
A. an = (an-1)5 with a1 = 3
B. an = an-1 - 5 with a1 = 3
C. an = 8/3·an-1 with a1 = 3
D. an = an-1 + 5 with a1 = 3
Patterns
7. What is the recursive formula for this sequence?
33 , 26 , 12 , -9 , -37 , ...
A.
a1 = 33, an = 7(an-1) - (n-1)
B.
a1 = 33, an = an-1 - 7(n)
C.
a1 = 33, an = an-1 - 7(n-1)
D.
a1 = 33, an = an-1 - 7
Simplify Expressions
8. Simplify.
A.
B.
C.
D.
Linear Equations
9. Solve for x.
A. x = 9
B. x = 8
C. x = 7
D. x = 6
Linear Equations
10. Solve for x.
A. x = -4
B. x = 5
C. x = -6
D. x = -5
Systems of Equations & Inequalities
11. The following system of equations is graphed below.
-2x - 2y = 0
-5x + 2y = 0
Find the solution to the system.
A. x = 0, y = 1
B. x = 1, y = 0
C. x = -5, y = 0
D. x = 0, y = 0
Simplify Expressions
12. Simplify the following expression.
3a3 - 4ab + 3a3
A. 6a6 - 4ab
B. 6a3 - 4ab
C. 9a3 - 4ab
D. 2a3 - 4b
Systems of Equations & Inequalities
13. Solve for x in the two equations below using substitution.
A.
B.
C.
D.
Linear Equations
14. Which algebraic expression is equivalent to the expression below?
8(4x + 4) + 2x
A. 34x + 32
B. 34x + 4
C. 6x + 32
D. 32x + 32
Patterns
15. What is the recursive formula for this sequence?
42 , 38 , 30 , 18 , 2 , ...
A.
a1 = 42, an = an-1 - 4(n-1)
B.
a1 = 42, an = an-1 - 4
C.
a1 = 42, an = an-1 - 4(n)
D.
a1 = 42, an = 4(an-1) - (n-1)
Systems of Equations & Inequalities
16. The side of a square is 8 inches longer than the side of an equilateral triangle. The perimeter
of the square is 48 inches more than the perimeter of the triangle. What are the lengths of a side
of the square, s, and a side of the triangle, t?
A. s = 8 inches; t = 16 inches
B. s = 24 inches; t = 32 inches
C. s = 40 inches; t = 32 inches
D. s = 24 inches; t = 16 inches
Quadratic Equations
17. Which algebraic expression is equivalent to the expression below?
(x - 4)2 - (x + 4)2
A. -16x
B. 2x2 + 32
C. 16x + 20
D. -2x2 - 16x - 32
Simplify Expressions
18. Simplify.
(4x - 2y)2 + 16xy
A. 16x2 + 4y2
B. 16x2 - 16xy + 4y2
C. 16x2 + 16xy + 4y2
D. 16x2 - 4y2
Quadratic Equations
19. Solve for x in the following equation.
x2 + 3x = 28
A. x = 4; -8
B. x = -4; 7
C. x = 4; -7
D. x = -1; 28
Systems of Equations & Inequalities
20. The following system of equations is graphed below.
2x - 3y = 13
5x + 5y = -5
Find the solution to the system.
A. x = 2, y = 3
B. x = 2, y = -3
C. x = -2, y = 3
D. x = -2, y = -3
Systems of Equations & Inequalities
21. Solve for y in the two equations below using substitution.
A.
B.
C.
D.
Simplify Expressions
22. Simplify the following expression.
3(5y + 3z) - 9z
A. 15y - 12z
B. 8y
C. 8y - 18z
D. 15y
Linear Equations
23. Solve for x.
A. x = 78
B. x = -15
C. x = 52
D. x = 65
Quadratic Equations
24. Solve for x in the following equation.
A.
B.
C.
D.
Systems of Equations & Inequalities
25. Find the solution to the system of equations given below using elimination by addition.
4x + 16y = 80
x + 2y = 10
A.
B.
C.
D.
x = 0, y = 5
x = -30, y = 20
x = 20, y = -5
x = 50, y = -20
Systems of Equations & Inequalities
26.
Which set of inequalities best represents the graph shown above?
y > -1/2x + 5
x > -2
A.
y > -2
y > -1/2x + 5
x > -2
B.
y > -2
C.
y < -1/2x + 5
x > -2
y > -2
y < -1/2x + 5
x > -2
D.
y > -2
Linear Equations
27. Solve for x.
A. x = 3
B. x = 2
C. x = -2
D. x = 4
Simplify Expressions
28. (2x4y2z)(6x3y2z)
A. 8x12y4z
B. 12x12y4z
C. 12x7y4z2
D. 8x7y4z2
Quadratic Equations
29. Which algebraic expression is equivalent to the expression below?
-3(-x + 4)2
A. 3x2 - 24x + 48
B. -3x2 + 24x - 48
C. -3x2 - 48
D. x2 - 8x + 16
Simplify Expressions
30. Simplify.
A.
B.
C.
D.
Patterns
31. Find the fifth term in the recursively defined sequence below if a1 = 1.
an = 2·an-1 + 3
A. 29
B. 13
C. 125
D. 61
Quadratic Equations
32. Solve for x in the following equation.
x2 + 10x + 24 = 0
A. x = -1; -24
B. x = -4; -6
C. x = 4; 6
D. x = 4; -6
Patterns
33. The first five terms of a sequence are given below.
34 , 36 , 38 , 40 , 42 , ...
Determine which of the following formulas gives the nth term of this sequence.
A. 32 + 2n
B. 33 + n
C. 35 - n
D. 36 - 2n
Simplify Expressions
34. Simplify.
5(4a - 5b) + 2(2a - 2b)
A. 24a - 29b
B. 24a - 21b
C. 42a - 49b
D. 24a - 7b
Patterns
35. What is the explicit formula for this sequence?
11 , 0 , -11 , -22 , -33 , ...
A. an = 22 - 11n
B. an = -11 + 22n
C. an = -11 - 22n
D. an = 22 - 10n
Linear Equations
36. Which value for x makes the sentence true?
9x + 4 = 22
A. x = 13
B. x = 4
C. x = 2
D. x = 9
Linear Equations
37. Solve for x.
A. x = -7
B. x = 8
C. x = -9
D. x = -8
Quadratic Equations
38. Which algebraic expression is equivalent to the expression below?
[-2x - 3 - (3x - 6)]2
A. x2 + 6x + 9
B. 25x2 + 90x + 81
C. x2 - 18x + 81
D. 25x2 - 30x + 9
Systems of Equations & Inequalities
39. Find the solution to the system of equations given below using elimination by addition.
A.
B.
C.
D.
Quadratic Equations
40. Solve for x in the following equation.
x(x - 10) = 11
A. x = 1; 11
B. x = -1; -11
C. x = 1; -11
D. x = -1; 11
Answers
1. A
2. B
3. A
4. B
5. A
6. D
7. C
8. B
9. B
10. D
11. D
12. B
13. B
14. A
15. A
16. D
17. A
18. A
19. C
20. B
21. A
22. D
23. D
24. A
25. A
26. D
27. B
28. C
29. B
30. B
31. D
32. B
33. A
34. A
35. A
36. C
37. D
38. D
39. A
40. D
Explanations
1.
2. Each number in the sequence is 3 more than the previous number (3n).
Use the first term to develop the nth term.
45 = ___ + 3(1)
45 = 42 + 3
an = 42 + 3n
3. Each number in the sequence is 8 more than the previous number (8n).
Use the first term to develop the nth term.
-4 = ___ + 8(1)
-4 = -12 + 8
an = -12 + 8n
4. Factor the equation and solve for x.
5. Given (1/5) (2x + 7) = 4, we are trying to solve for, or isolate, x the simplest way possible. We
would start by either distributing the 1/5 or multiplying by 5. Staying away from fractions is
usually the simplest way. So, to get rid of the 1/5, multiply each side by 5.
6. Find the first four or five numbers in the sequence to identify a pattern.
a1 = 5(1) - 2 = 3
a2 = 5(2) - 2 = 8
a3 = 5(3) - 2 = 13
a4 = 5(4) - 2 = 18
a5 = 5(5) - 2 = 23
Every term is 5 more than the previous term. So,
an = an-1 + 5
would be the recursive definition of the given sequence.
7. When finding a recursive formula, examine how each term differs from the previous term.
This sequence decreases by multiples of 7: first by 7, then 14, then 21, then 28,...
Each term (an) is the value of the previous term (an-1) decreased by 7 times the number of the
previous term (n-1).
For instance, 26 = 33 - (7 × 1).
In other words,
current term = previous term - (7 × previous term number).
Now, write it as a recursive formula.
a1 = 33, an = an-1 - 7(n-1)
8. When an exponential expression is raised to another exponent, multiply the exponents.
When dividing exponential expressions with the same base, subtract the exponents.
Any number raised to the zero power is equal to 1.
9.
10.
11. The point of intersection of the two graphs gives the solution to the system of equations.
The two lines intersect at (0, 0).
So, x = 0 and y = 0 is the solution.
12. Simplify the expression by combining like terms.
3a3 - 4ab + 3a3 = 6a3 - 4ab
13. Solve the first equation for y.
Now, substitute this answer for y into the second equation.
Then, solve for x.
14. Use properties of arithmetic to transform the given expression.
8(4x + 4) + 2x
32x + 32 + 2x Distribute the 8.
Combine like terms.
34x + 32
15. When finding a recursive formula, examine how each term differs from the previous term.
This sequence decreases by multiples of 4: first by 4, then 8, then 12, then 16,...
Each term (an) is the value of the previous term (an-1) decreased by 4 times the number of the
previous term (n-1).
For instance, 38 = 42 - (4 × 1).
In other words,
current term = previous term - (4 × previous term number).
Now, write it as a recursive formula.
a1 = 42, an = an-1 - 4(n-1)
16. The side of the square is 8 inches longer than the side of the equilateral triangle.
s-t=8
The perimeter of the square is 48 inches more than the perimeter of the triangle.
4s - 3t = 48
To find the side lengths of the square and triangle, solve the system of equations below.
s-t=8
4s - 3t = 48
In this case, solve by substitution.
s-t =8
s=t+8
So,
4s - 3t = 48
4(t + 8) - 3t = 48
4t + 32 - 3t = 48
t = 16
and
s=t+8
s = 16 + 8
s = 24.
Therefore, the length of a side of the square is 24 inches, and the length of a side of the triangle
is 16 inches.
17. Use properties of arithmetic to transform the given expression into one of the answer choices.
(x - 4)2 - (x + 4)2
(x2 - 8x + 16) - (x2 + 8x + 16)
-16x
18.
19. Factor the equation and solve for x.
x2 + 3x = 28
x2 + 3x - 28 = 0
(x - 4)(x + 7) = 0
x = 4; -7
20. The point of intersection of the two graphs gives the solution to the system of equations.
The two lines intersect at (2, -3).
So, x = 2 and y = -3 is the solution.
21. Solve the first equation for x.
Now, substitute this answer for x into the second equation.
Then, solve for y.
22. Simplify the expression by using the distributive property and combining like terms.
3(5y + 3z) - 9z = 15y + 9z - 9z
= 15y
23. Keep in mind that the goal is to get x by itself.
24. Factor the equation and solve for x.
25. Use elimination by addition to solve the system of equations.
4x + 16y = 80
x + 2y = 10
Start by eliminating the x term. To do this, multiply the second equation by -4, and then add the
two equations together.
4x + 16y = 80
-4x - 8y = -40
8y = 40
y= 5
Next, substitute y = 5 into the original second equation, and solve for x.
x + 2y = 10
x + 2(5) = 10
x + 10 = 10
x = 10 - 10
x=0
Therefore, the solution to the system of equations is x = 0, y = 5.
26. First look at the slanted line. It follows the path of y = -1/2x + 5. Notice that the shaded region
is below the line and the line is solid. Therefore, the correct inequality for this part of the graph
is y < -1/2x + 5.
Next look at the vertical line. It follows the path of x = -2. Notice that the shaded region is to the
right of the line and the line is solid. Therefore, the correct inequality for this part of the graph is
x > -2.
Finally, look at the horizontal line. It follows the path of y = -2. Notice that the shaded region is
above the line and the line is dotted. Therefore, the correct inequality for this part of the graph is
y > -2.
27.
28. When multiplying exponential expressions with the same base, add the exponents. So, break
up the given terms into terms with common bases.
(2x4y2z)(6x3y2z)
= (2)(6)(x4)(x3)(y2)(y2)(z)(z)
= 12x(4+3)y(2+2)z(1+1)
= 12x7y4z2
29. Use properties of arithmetic to transform the given expression into one of the answer choices.
-3(-x + 4)2
-3(x2 - 8x + 16)
-3x2 + 24x - 48
30. When an exponential expression is raised to another exponent, multiply the exponents on
each term.
31. a1 = 1
a2 = 2·a1 + 3 = 2·1 + 3 = 2 + 3 = 5
a3 = 2·a2 + 3 = 2·5 + 3 = 10 + 3 = 13
a4 = 2·a3 + 3 = 2·13 + 3 = 26 + 3 = 29
a5 = 2·a4 + 3 = 2·29 + 3 = 58 + 3 = 61
So, a5 = 61.
32. Factor the equation and solve for x.
x2 + 10x + 24 = 0
(x + 4)(x + 6) = 0
x = -4; -6
33. A generic arithmetic sequence is of the following form,
a , a + d , a + 2d , a + 3d , ... , a + (n - 1)d , ...
where a is the first term, d is the common difference, and a + (n - 1)d is the nth term.
In this case, the first term is 34 and the common difference is 2.
Therefore, the nth term is as follows.
nth term = a + (n - 1)d
= 34 + (n - 1)(2)
= 34 + (2n - 2)
= 32 + 2n
34.
35. Each number in the sequence is 11 less than the previous number (11n).
Use the first term to develop the nth term.
11 = ___ - 11(1)
11 = 22 - 11
an = 22 - 11n
36. Keep in mind that the goal is to get x by itself.
9x + 4
9x + 4 - 4
9x
9x ÷ 9
x
=
=
=
=
=
22
22 - 4
18
18 ÷ 9
2
37.
38. Use properties of arithmetic to transform the given expression into one of the answer choices.
[-2x - 3 - (3x - 6)]2
(-5x + 3)2
25x2 - 30x + 9
39. Use elimination by addition to solve the system of equations.
Start by eliminating the y term. To do this, multiply the first equation by -3, and then add the two
equations together.
Next, substitute x = -5 into the first equation, and solve for y.
Therefore, the solution to the system of equations is
40. Factor the equation and solve for x.
.
x(x - 10) = 11
x2 - 10x = 11
x2 - 10x - 11 = 0
(x + 1)(x - 11) = 0
x = -1; 11