Definitions
System of Equations
2 or more equations that are related.
x + y = 6
2x + 4y = 18
You could probably solve another way and we wouldn't know. But good sportsmanship? :)
x = 3
y = 3
Solve the following system using the elimination method.
3x+2y=11
-x+y=3
Multiply the second equation by 2 to eliminate y. This leaves -2x+2y=6.
Subtract -2x+2y=6 from the first equation 3x+2y=11. This leaves 5x=5. Divide by 5 on both sides, making x=1.
Substitute x=1 into either equation. -1+y=3. y=4. The solution is (1,4).
William, Xing Mei, Yuki, and Zack run a race. In how many different ways can they finish?
(A) 4 (B) 16 (C) 24 (D) 32 (E) 64
There are 4! ways to finish the race.
4! = 4 โ 3 โ 2 โ 1
= 24
A recipe calls for 4 parts sugar per 11 parts flour.
How many cups of sugar are needed for 132 cups of flour?
Set up a ratio with sugar in the numerator and flour in the denominator. 4/11=x/132.
Cross multiply. 4*132=11x. 4*132=528. 11x=528. Divide by 11 on both sides. x=48.
48 cups of sugar are needed for the recipe.
Elimination
Adding or Subtracting 2 equations to remove a variable.
Solve the following system using the substitution method.
2y=12x+8
3x+4y=70
Divide the first equation by 2 in order to solve for y. y=6x+4.
Substitute 6x+4 into y for the second equation.
3x+4(6x+4)=120. 3x+24x+16=120. 27x+16=70.
27x=54. Divide by 27 on both sides. x=2.
Plug 2 into x for y=6x+4. 6(2)+4=16, so y=16. The solution is (2,16).
Solve the following system using the elimination method.
4x+3y=24
12x+5y=100
Multiply the first equation by 3 to eliminate x, resulting in 12x+9y=72.
Subtract the second equation from the first, leaving 4y=-28. Divide by 4 on both sides to get y=-7.
Plug -7 into y for either equation. 4x+3(-7)=24. 4x-21=24. Add 21 to both sides. 4x=45. Divide by 4 on both sides. x=45/4. The solution is (45/4,-7).
Phone Company A charges 50 + 3x dollars for an international phone plan, where x is the number of minutes spent talking. Phone Company B charges 60 + 2x dollars for an international phone plan, where x is the number of minutes spent talking. What is the price at which both companies charges the same amount?
(A) $10 (B) $20 (C) $30 (D) $80 (E) $110
Charge of company A = 50 + 3x
Charge of company B = 60 + 2x
50 + 3x = 60 + 2x
3x - 2x = 60 - 50
x = 10 (Number of minutes)
Charge of company A = 50 + 3(10)
= 50 + 30
= $80
Hence the price is $80.
Chad and George are painting a room. Chad can finish the job alone in 8 hours, while George can finish the job alone in 10 hours. How long, in hours, would it take for both to finish the job at the same time without breaks?
(A) 9/40 (B) 4 (C) 40/9 (D) 37/8
Chad can finish 1/8 of the job in an hour.
George can finish 1/10 of the job in an hour.
Together, their rate would be 1/8+1/10 of the job per hour. 1/8+1/10=9/40. This means that they can complete 9/40 of the job per hour.
Divide 1 job by 9/40 of the job per hour to find the number of hours. The job would take 40/9 hours.
Substitution
Substituting the 2 equations into each other in order to remove a variable
Which of the following must be the value of x such that 2x/3+2y=6 and y+16/y=4?
(A) 0 (B) -2 (C) 3 (D) -3
Multiply the first equation by 3 to get rid of the fraction. 2x+6y=18.
Multiply the second equation by y to get rid of the y in the denominator. y^2+16=4y. To make the right side of the equation equal to 0, subtract 3y on both sides. y^2-4y+16=0.
Factor the equation. (y-4)^2=0. y must equal 4.
Substitute y=4 into the first equation. 2x+6(4)=18.
2x+24=18. Subtract 24 on both sides. 2x=-6. Divide by 2 on both sides. x=-3.
Solve the following system using the elimination method.
10x+3y=500
60x+18y=3000
If you divide the second equation by 6, you are left with the first equation. Therefore, there are infinitely many solutions as long as x=-3y/10+50 and y=-10x/3+500/3.
Lenny's average score after 3 tests is 88. What score on the 4th test would bring Lenny's average upto exactly 90?
(A) 92 (B) 93 (C) 94 (D) 95 (E) 96
Sum of 3 test scores = 3 (88) = 264
Sum of 4 test scores = 4(90) = 360
4th test score = 360 - 264
= 96
In the synthesis reaction for NH3, 1 mole of N2 and 3 moles of H2 are needed for every 2 moles of NH3. If the reaction goes to completion with 1.5 moles of N2, how many moles of H2 were used in the reaction?
The amount of moles of NH3 is irrelevant. Given that 1 mole of N2 is needed for every 3 moles of H2, the amount of H2 is 3 times the amount of H2. Therefore, 4.5 moles of H2 were used because 1.5 times 3 is 4.5.
Ratio
The relationship between parts of one thing to parts of another thing.
Which of the following values of a would result in no solutions for the system
9x-14y=-3, 2x-ay=-6
(A) -9/14 (B) -28/9 (C) 9/14 (D) 28/9
A linear system of equations with no solutions would result in two distinct lines with the same slope.
If you put both equations into slope-intercept form, you get y=9/14x+3/14 and y=2/ax+6/a. If the equations need the same slope, 9/14=2/a. Multiply both sides by a. 9a/14=2. 2รท9/14=2*14/9. a=28/9.
Solve for x using the elimination method.
3csc(3๐/2)+8x=45
6csc(3๐/2)+21x=120
Assume that csc(3๐/2) is some variable y. This leaves us with 3y+8x=45 and 6y+21x=120.
Multiply the first equation by 2 to eliminate y. This gives us 6y+16x=90.
Subtract 6y+16x=90 from 6y+21x=120, giving us 5x=30. Divide by 5 on both sides. x=6.
William, Xing Mei, Yuki, and Zack run a race. In how many different ways can they finish?
(A) 4 (B) 16 (C) 24 (D) 32 (E) 64
There are 4! ways to finish the race.
4! = 4 โ 3 โ 2 โ 1
= 24
Karl and Dave are construction workers. Dave can finish a particular job in 10 hours. Together, they can finish that same job in 4 hours. How long, in hours, would it take Karl to finish the job alone?
(A) 20/3 (B) 6 (C) 2/9 (D) 3/20
If Dave can finish the job in 10 hours, he can finish 1/10 of the job per hour.
If both workers can finish the job in 4 hours, they would finish 1/4 of the job in an hour.
To find Karl's rate, we can subtract Dave's rate from the collective rate. 1/4-1/10=3/20.
Karl can finish 3/20 of the job in one hour. One job divided by 3/20 of the job per hour gets us 20/3 hours.
Rate
The amount of something per amount something else
Let (a,b) be some point in the coordinate plane such that yโค -15x+3000 and yโค5x. What is the maximum possible value of b?
Substitute 5x for y. 5xโค -15x+3000.
Add 15x to both sides. 20xโค3000.
Divide by 20 on both sides. xโค3000.
yโค5x, so yโค5(150).
yโค750, so b is at most 750.
A total of $12,000 is invested in two funds paying 9% and 11% simple interest. If the yearly interest is $1,180, how much of the $12,000 is invested at each rate?
Let x be the amount invested into the fund paying 9% interest and y be the amount invested into the fund paying 11% interest.
Given this, x+y=12000 and 0.09x+0.11y=1180.
Multiply the first equation by 0.09 to eliminate x. This gives us 0.09x+0.09y=1080. Subtract this from the second equation, giving us 0.02y=100. Divide both sides by 0.02. y=5000.
Substitute 5000 into y for the first equation. 12000-5000=7000. Thus, $5,000 was invested at 9% interest, and $7,000 was invested at 11% interest.
Darius ran 1/6 as many times around the track as Ezekiel. Darius ran around the track 2 โ times. How many times did Ezekiel run around the track?
(A) 8 (B) 10 (C) 12 (D) 14 (E) 16
Let "x" be the number of times that Ezekiel around the track, then
Darius ran around the track = (1/6)x = x/6
Now, Darius ran around the track = 2 โ times
= 8/3
We need to show 8/3 in terms of 1/6.
For that, let us multiply the numerator and denominator by 2.
So, we get
= (8/3) โ (2/2)
= 16/6
= (1/6) โ 16
Hence Ezekiel run around the track 16 times.
In a box of blocks, there are 3 red blocks for every 2 green blocks. There are 5 blue blocks for every 2 red blocks and 7 yellow blocks for every green block. If there are 56 yellow blocks in the box, how many blue blocks are in the box?
Red:green - 3:2, Blue:red - 5:2, Yellow:green - 7:1
Yellow:green - 14:2, Red:yellow - 3:14,
Blue:red - 15:6, Red:yellow - 6:28
Yellow:blue - 28:15 = 56:30
There are 30 blue blocks.