algebra
The line shown by the equation 3x + 2y = 18 has what y-intercept?
9
Set x = 0:
2y = 18, so y = 9.
The function f(x) = 2(x - 5)2 + 3 has vertex (h, k). What is h + k?
8
Vertex form is a(x - h)^2 + k.
So the vertex is (5, 3).
5 + 3 = 8.
A recipe uses 3 cups of flour for every 5 cups of sugar. If 20 cups of sugar are used, how many cups of flour are needed?
12
3/5 = x/20
5x = 60
x = 12.
In isosceles triangle ABC, AB = AC and angle B = 47°. What is the measure of angle A?
86 degrees
Since AB = AC, the base angles are equal, so angle B = angle C = 47°.
Triangle angles add to 180°:
47 + 47 + A = 180
94 + A = 180
A = 86°
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If one marble is chosen at random, what is the probability that it is not blue?
7/10
Total marbles:
5 + 3 + 2 = 10
Not blue means red or green:
5 + 2 = 7
So the probability is:
7/10
For what value of k does the equation below have no solution?: 4x + 7 = 4x + k
Any value except 7.
Subtract 4x from both sides:
7 = k.
If k = 7, there are infinitely many solutions.
If k ≠ 7, there is no solution.
The quadratic function g(x) = (x - 4)(x + 7) has two x-intercepts. What is the positive x-intercept?
4
Set each factor equal to 0:
x - 4 = 0 → x = 4
x + 7 = 0 → x = -7
The positive x-intercept is 4.
A jacket originally costs $80. It is discounted by 25%. What is the sale price?
$60
A 25% discount means the customer pays 75%.
0.75(80) = 60.
In right triangle XYZ, legs XY = 9 and YZ = 12, with a right angle at Y. What is the length of XZ?
15
XZ is the hypotenuse. Use the Pythagorean theorem:
9² + 12² = XZ²
81 + 144 = XZ²
225 = XZ²
XZ = 15
A car travels at a speed of 54 miles per hour. How many feet per second is this?
Use 1 mile = 5,280 feet.
79.2
Convert miles per hour to feet per second:
54 miles/hour × 5,280 feet/mile = 285,120 feet/hour
There are 3,600 seconds in an hour:
285,120 / 3,600 = 79.2
So the speed is 79.2 feet per second.
The system below has solution (x, y). What is x + y?
2x + 3y = 19
5x - 3y = 16
8
Add the equations:
7x = 35, so x = 5.
Then 2(5) + 3y = 19, so 3y = 9, and y = 3.
x + y = 8.
Which expression is equivalent to (3x4y2)(2x3y5)?
6x7y7
Multiply coefficients: 3 · 2 = 6.
Add exponents with the same base:
x^4 · x^3 = x^7
y^2 · y^5 = y^7.
The mean of the numbers 6, 8, 10, 12, and x is 11. What is x?
19
Mean = sum ÷ count.
Total needed: 11 · 5 = 55.
Known total: 6 + 8 + 10 + 12 = 36.
x = 55 - 36 = 19.
In triangle PQR, an exterior angle at R measures 127°. One remote interior angle, angle P, measures 52°. What is the measure of angle Q?
75 degrees
An exterior angle equals the sum of the two remote interior angles:
127 = 52 + Q
Q = 75°
The equation of a circle is shown below:
(x - 4)2 + (y + 3)2 = 49
What is the radius of the circle?
7
The standard equation of a circle is:
(x - h)2 + (y - k)2 = r2
Here:
r2 = 49
So:
r = 7
For what value of a will the lines below be parallel but not identical?
ax + 6y = 12
4x + 3y = 9
8
Rewrite the second line:
4x + 3y = 9 has slope -4/3.
First line:
ax + 6y = 12 → 6y = -ax + 12 → slope -a/6.
Set slopes equal:
-a/6 = -4/3
a/6 = 4/3
a = 8.
If h(t) = 120(0.85)t, what does the number 0.85 represent? (be detailed)
The quantity is multiplied by 0.85 each time t increases by 1, so it decreases by 15% each time.
0.85 = 1 - 0.15, so it represents 15% decay per unit of time.
A data set consists of 5 consecutive integers. When the number 44 is added to the data set, the mean increases by 5. What is the greatest of the original 5 integers?
16
Let the original integers be:
x, x + 1, x + 2, x + 3, x + 4
Original mean:
x + 2
After adding 44, the new mean is:
x + 2 + 5 = x + 7
Set up the new mean:
(x + x + 1 + x + 2 + x + 3 + x + 4 + 44) / 6 = x + 7
(5x + 54) / 6 = x + 7
5x + 54 = 6x + 42
x = 12
Original integers: 12, 13, 14, 15, 16
In right triangle LMN, angle M = 90°, angle L = 30°, and side MN = 7. Find the exact value of LN + LM.
14 + 7√3
This is a 30-60-90 triangle. The side opposite 30° is the short leg.
Since angle L = 30°, the side opposite it is MN, so:
MN = 7
In a 30-60-90 triangle, the side ratio is:
short leg : long leg : hypotenuse = 1 : √3 : 2
So:
LN = 14
LM = 7√3
Therefore:
LN + LM = 14 + 7√3
In triangle ABC, points D and E lie on AB and AC, and DE is parallel to BC. If AD = 6, DB = 3, and AE = 8, what is AC?
12
Since DE || BC, triangles ADE and ABC are similar.
First find AB:
AB = AD + DB = 6 + 3 = 9
Set up the similarity ratio:
AD / AB = AE / AC
6 / 9 = 8 / AC
Simplify:
2 / 3 = 8 / AC
Cross multiply:
2AC = 24
AC = 12
A gym charges a one-time registration fee plus a monthly fee. After 4 months, the total cost is $164. After 9 months, the total cost is $329. If C = mx + b, where x is the number of months, what is the registration fee?
$32
Use the two points: (4, 164) and (9, 329).
Slope/monthly fee:
(329 - 164)/(9 - 4) = 165/5 = 33.
So C = 33x + b.
Use (4, 164):
164 = 33(4) + b
164 = 132 + b
b = 32.
The quadratic equation x2 - 10x + c = 0 has exactly one real solution. What is the value of c?
25
A quadratic has exactly one real solution when the discriminant equals 0.
For ax2 + bx + c = 0:
b2 - 4ac = 0.
Here, a = 1, b = -10, and c = c.
(-10)2 - 4(1)(c) = 0
100 - 4c = 0
c = 25.
Two box-and-whisker plots summarize the test scores for Class A and Class B.
Class A:
Minimum = 42, Q1 = 58, Median = 72, Q3 = 88, Maximum = 96
Class B:
Minimum = 50, Q1 = 64, Median = 72, Q3 = 80, Maximum = 100
Which statement must be true?
A) Class A has a greater interquartile range than Class B.
B) Class B has a greater range than Class A.
C) Class A has a greater mean than Class B.
D) At least half of Class B scored 80 or higher.
A
IQR = Q3 − Q1.
Class A IQR:
88 - 58 = 30
Class B IQR:
80 - 64 = 16
So Class A has the greater IQR.
Why the others are wrong:
Class A range = 96 - 42 = 54
Class B range = 100 - 50 = 50, so B is false.
Box plots do not show the mean, so C cannot be determined.
Q3 = 80 means about 75% scored at or below 80, not that half scored 80 or higher.
In right triangle ABC, angle C = 90°. If tan A = 3/4 and hypotenuse AB = 50, what is the area of triangle ABC?
600
For angle A:
tan A = opposite / adjacent = 3/4
So the legs are in a 3:4 ratio. A right triangle with legs in a 3:4 ratio has hypotenuse ratio 5, making it a 3-4-5 triangle.
Since the hypotenuse is 50, the scale factor is:
50 / 5 = 10
So the legs are:
3 · 10 = 30
4 · 10 = 40
Area of a triangle:
A = 1/2 bh
A = 1/2(30)(40)
A = 600
The function f is defined by:
f(x) = x^2 - 6x + 11
The function g is defined by:
g(x) = f(x - 2)
What is the minimum value of g(x)?
2
First find the minimum value of f(x).
Rewrite:
f(x) = x^2 - 6x + 11
The vertex x-value is:
x = -b / 2a = -(-6) / 2(1) = 3
Now plug in x = 3:
f(3) = 3^2 - 6(3) + 11
f(3) = 9 - 18 + 11 = 2
The transformation g(x) = f(x - 2) shifts the graph horizontally, but it does not change the minimum value.
So the minimum value of g(x) is:
2