All ACT Math Resources
Example Question #1 : Systems Of Equations
What is the sum of the x, y, and z coordinates of the point that satisfies this system of equations:
2x + 3y + 4z = 20
x = y – z
–3x + 2y + 2z = 23
Multiplying the second equation by 2 and subtracting it from the first gives the equation:
5y + 2z = 20
Multiplying the first equation by 3, multiplying the third equation by 2, and then adding those equations gives:
13y + 16z = 106
Now we have a system of two equations and two unknowns.
Multiply the equation 5y + 2z = 20 by 8, and then subtract this from 13y + 16z = 106
This yields –27y = –54, so y = 2. Then using the equation 5y + 2z = 20, z = 5. Then using any of the original three equations, x = –3.
The point of intersection is (-3, 2, 5), and the sum of these coordinates is 4.
Example Question #2 : Systems Of Equations
The cost of a movie ticket and a candy bar is $5. The cost of two tickets and a candy bar is $8.75. How much is a candy bar?
We start by setting up a system of equations. The price of one ticket and one candy bar is $5, so t+c=5. The price of two tickets and one candy bar is $8.75, so 2t+c=8.75. We can use the first equation to find out that c=5-t. We then substitute that value into the second equation, giving us 2t+(5-t)=8.75. This simplifies to 2t-t+5=8.75, so t+5=8.75, and finally t=3.75. We use the final value for t in the first equation, so 3.75+c=5. We solve for c, and get c=1.25.
Example Question #3 : Systems Of Equations
Jacob is 3 years older than Sarah, and Caroline is twice as old as Sarah. If Caroline is 28 years old, how many years old is Jacob?
One can describe the ages of Jacob, Sarah, and Caroline with the letters J, S, and C, respectively. From the information in the problem, J = S + 3, and C = 2S. Since C = 28, S = 28/2 = 14, and J = 14 + 3 = 17. Jacob is 17 years old.
Example Question #1 : How To Find The Solution For A System Of Equations
If ab = 24, a + b = 10, and a < b, what is the value of a – b?
Solving the second equation for b and substituting b = 10 – a into the equation ab = 24 gives us
a(10 – a) = 10a – a2 = 24
which can be set up and solved as the following quadratic equation:
a1 – 10a +24 = 0
(a – 6) (a – 4) = 0
a = 6, a = 4
a must be 4 and b must be 6, since a < b; therefore, 4 – 6 = –2.
Example Question #5 : How To Find The Solution For A System Of Equations
Josh is counting his money. He has only quarters and nickels. He has two more quarters than nickels. He counts his money to discover he has $1.40. How many total coins does he have?
None of the answers are correct
The general formula for money problems is V1 x N1 + V2 x N2 = $Value where V is the value of the coin and N is the number of coins for each separate type of coin involved. $Value is the total value of all the money when counted. With this problem N = number of nickels and
Q = N + 2 is the number of quarters. Substituting into the general formula we get
0.25(N + 2) + 0.05N = 1.40. Solving for N yields N = 3, therefore Q = 5. So the answer to the question is actually N + Q = 3 + 5 = 8 total coins.
Example Question #5 : Systems Of Equations
A business makes $5 million less the second year than the first year. In the third year the business makes twice as much as in the second year. If the business makes $15 million in the third year how much did it make in the first?
None of the other answers
This problem is probably best solved by developing a series of equations. To relate the second and third years we can set up the equation 15 = 2x where x is the amount of money made in the second year, which is two times less than the amount of money made in the third. By dividing each side by two we see that the business made 7.5 million dollars in the second year. To relate the first and second years we can set up the equation x – 5 = 7.5 where x is the amount of money made in the first year. By adding 5 to both sides of the equation we are able to see that the business made 12.5 million dollars in the first year.
Example Question #6 : Systems Of Equations
If Sally makes $9.50 an hour, what was her gross pay for the week if she gets paid time-and-a-half for any hours above 40?
Sally works 40 regular hours and 5 overtime hours. Her regular hourly pay becomes 40 * 9.50 = $380.00, and her overtime pay becomes 5 * 9.50 * 1.5 = $71.25, so her weekly gross (before taxes) pay is $451.25
Example Question #7 : Systems Of Equations
Sally sells cars for a living. She has a monthly salary of $1,000 and a commission of $500 for each car sold. How much money would she make if she sold seven cars in a month?
The commission she gets for selling seven cars is $500 * 7 = $3,500 and added to the salary of $1,000 yields $4,500 for the month.
Example Question #8 : Systems Of Equations
John's age is half of Bob's age in 2010. John is also 3 years younger than Bob in 2010.
How old is Bob in 2013?
Using algebra, we can write the system of equations J = (1/2)B and J + 3 = B, where J is John's age in 2010 and B is Bob's age in 2010.
We plug in and get that J = 3 and B = 6
So 3 years later in 2013, we know Bob is 9 years old.
Example Question #9 : Systems Of Equations
Given the system of equations:
2x + 3y = 15
x + 3y = 0
What is x?
Substitute the second equation into the first to find that y = –5. Then plug y = –5 into either equation to find x = 15.