Systems of Equations - ACT Math
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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?
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.
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.
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Consider the following system of equations: x – y = 5 and 2x + y = 4.
What is the sum of x and y?
Consider the following system of equations: x – y = 5 and 2x + y = 4.
What is the sum of x and y?
Add the two equations to get 3x = 9, so x = 3. Substitute the value of x into one of the equations to find the value of y; therefore x = 3 and y = -2, so their sum is 1.
Add the two equations to get 3x = 9, so x = 3. Substitute the value of x into one of the equations to find the value of y; therefore x = 3 and y = -2, so their sum is 1.
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Joey is four years older than Billy. The sum of their ages is 24. How old is Billy?
Joey is four years older than Billy. The sum of their ages is 24. How old is Billy?
Define variables as x = Billy's age and x + 4 = Joey's age
The sum of their ages is x + (x + 4) = 24
Solving for x, we get that Billy is 10 years old and Joey is 14 years old.
Define variables as x = Billy's age and x + 4 = Joey's age
The sum of their ages is x + (x + 4) = 24
Solving for x, we get that Billy is 10 years old and Joey is 14 years old.
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Find a solution for the following system of equations:
x – y = 2
2_x_ – 3_y_ = 3
Find a solution for the following system of equations:
x – y = 2
2_x_ – 3_y_ = 3
Use substitution and solve for one variable, then back substitute and solve for the other variable, or use elimination.
Use substitution and solve for one variable, then back substitute and solve for the other variable, or use elimination.
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Find a solution to the system of equations:
2_x_ – y = 0
x + y = 3
Find a solution to the system of equations:
2_x_ – y = 0
x + y = 3
Use substitution and plug in to solve for one equation. Then use back substitution to solve for the other variable.
Use substitution and plug in to solve for one equation. Then use back substitution to solve for the other variable.
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What is the sum of the x, y, and z coordinates of the point that satisfies this system of equations:
2_x_ + 3_y_ + 4_z_ = 20
x = y – z
–3_x_ + 2_y_ + 2_z_ = 23
What is the sum of the x, y, and z coordinates of the point that satisfies this system of equations:
2_x_ + 3_y_ + 4_z_ = 20
x = y – z
–3_x_ + 2_y_ + 2_z_ = 23
Multiplying the second equation by 2 and subtracting it from the first gives the equation:
5_y_ + 2_z_ = 20
Multiplying the first equation by 3, multiplying the third equation by 2, and then adding those equations gives:
13_y_ + 16_z_ = 106
Now we have a system of two equations and two unknowns.
Multiply the equation 5_y_ + 2_z_ = 20 by 8, and then subtract this from 13_y_ + 16_z_ = 106
This yields –27_y_ = –54, so y = 2. Then using the equation 5_y_ + 2_z_ = 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.
Multiplying the second equation by 2 and subtracting it from the first gives the equation:
5_y_ + 2_z_ = 20
Multiplying the first equation by 3, multiplying the third equation by 2, and then adding those equations gives:
13_y_ + 16_z_ = 106
Now we have a system of two equations and two unknowns.
Multiply the equation 5_y_ + 2_z_ = 20 by 8, and then subtract this from 13_y_ + 16_z_ = 106
This yields –27_y_ = –54, so y = 2. Then using the equation 5_y_ + 2_z_ = 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.
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Solve for x based on the following system of equations:
x + y = 5
2x + 3y = 20
Solve for x based on the following system of equations:
x + y = 5
2x + 3y = 20
One method of solving a system of equations requires multiplying one equation by a factor that will allow for the removal of one variable. In this system, we can multiply (x+y=5) by -2. When the -2 is distributed across the entire equation, the equation becomes (-2x-2y=-10). We then add the two equations: (-2x-2y=-10) + (2x+3y=20). When we do this, the x variable cancels out, leaving us with y=10. We then subsitute 10 for y in either of the original equations: (x+10=5) or (2x+ 30=20). Either way, we end up with x=-5. If you got an answer of 5, you may have made a computation error. If you got 10, you may have forgotten to substitute the y-value into one of the original equations to solve for x.
One method of solving a system of equations requires multiplying one equation by a factor that will allow for the removal of one variable. In this system, we can multiply (x+y=5) by -2. When the -2 is distributed across the entire equation, the equation becomes (-2x-2y=-10). We then add the two equations: (-2x-2y=-10) + (2x+3y=20). When we do this, the x variable cancels out, leaving us with y=10. We then subsitute 10 for y in either of the original equations: (x+10=5) or (2x+ 30=20). Either way, we end up with x=-5. If you got an answer of 5, you may have made a computation error. If you got 10, you may have forgotten to substitute the y-value into one of the original equations to solve for x.
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If st + 12= 3s + sv, and t – v = 7, what is the value of s?
If st + 12= 3s + sv, and t – v = 7, what is the value of s?
We are given st + 12= 3s + sv and t-v=7. Since we have a numerical value for t-v+, it would be ideal to isolate that in the first equation. We can do this be rearranging the first equation. Starting with st + 12= 3s + sv, we can subtract the 12 from the left giving us a -12 on the right. We then subtract the sv from the right and put it on the left. This gives us st-sv=3s-12. We then factor out the s on the left side of the equation, giving us s(t-v)=3s-12. We then plug-in 7 for t-v, giving us 7s=3s-12. We then solve for s, giving us -3.
We are given st + 12= 3s + sv and t-v=7. Since we have a numerical value for t-v+, it would be ideal to isolate that in the first equation. We can do this be rearranging the first equation. Starting with st + 12= 3s + sv, we can subtract the 12 from the left giving us a -12 on the right. We then subtract the sv from the right and put it on the left. This gives us st-sv=3s-12. We then factor out the s on the left side of the equation, giving us s(t-v)=3s-12. We then plug-in 7 for t-v, giving us 7s=3s-12. We then solve for s, giving us -3.
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If
and
Which of the following expresses 
in terms of 
?
If
and
Which of the following expresses
First we must solve for 
, then substitute into the other equation. Since we want 
in terms of 
, solve for 
in the 
equation and substitute our value of 
(in terms of 
) into the 
equation, then simplify:
Now that we have 
, let's plug that into the 
equation.
Already we can see that this problem is a mess because it is an expression with two denominators. Remember that dividing by a number is equal to multiplying by that number's inverse. Thus, dividing by 
is the same as multiplying by 
. So let's make an equivalent expression look like this:
This is much better as we can multiply straight across to get:
Now we can solve for 
.
First we must solve for
Now that we have
Already we can see that this problem is a mess because it is an expression with two denominators. Remember that dividing by a number is equal to multiplying by that number's inverse. Thus, dividing by
This is much better as we can multiply straight across to get:
Now we can solve for
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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?
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.
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.
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If ab = 24, a + b = 10, and a < b, what is the value of a – b?
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) = 10_a_ – _a_2 = 24
which can be set up and solved as the following quadratic equation:
a_1 – 10_a +24 = 0
(a – 6) (a – 4) = 0
a = 6, a = 4
a must be 4 and b must be 6, since a < _b; t_herefore, 4 – 6 = –2.
Solving the second equation for b and substituting b = 10 – a into the equation ab = 24 gives us
a(10 – a) = 10_a_ – _a_2 = 24
which can be set up and solved as the following quadratic equation:
a_1 – 10_a +24 = 0
(a – 6) (a – 4) = 0
a = 6, a = 4
a must be 4 and b must be 6, since a < _b; t_herefore, 4 – 6 = –2.
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For what negative value of n does the system of equations yield no solutions?
3x + ny = 17
nx + (n + 6)y = 7
For what negative value of n does the system of equations yield no solutions?
3x + ny = 17
nx + (n + 6)y = 7
Plugging in **–**3 for n gives a system that, when added vertically, gives 0 = 24, which is untrue.
Plugging in **–**3 for n gives a system that, when added vertically, gives 0 = 24, which is untrue.
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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?
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?
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.
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.
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How much pure water should be added to 1 gallon of pure cleaning solution to dilute it to 60% strength by volume?
How much pure water should be added to 1 gallon of pure cleaning solution to dilute it to 60% strength by volume?
Pure water is considered 0% whereas pure solution is 100%.
The general equations is Vwater x Pwater + Vsoultion x Psolution = Vfinal x Pfinal where
V means volume and P means percent.
x(0) + 1(1.00) = (x + 1)(0.60) and solve for x = volume of pure water.
Pure water is considered 0% whereas pure solution is 100%.
The general equations is Vwater x Pwater + Vsoultion x Psolution = Vfinal x Pfinal where
V means volume and P means percent.
x(0) + 1(1.00) = (x + 1)(0.60) and solve for x = volume of pure water.
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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?
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?
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.
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.
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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?
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
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
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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?
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.
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.
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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?
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.
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.
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Given the system of equations:
2_x_ + 3_y_ = 15
x + 3_y_ = 0
What is x?
Given the system of equations:
2_x_ + 3_y_ = 15
x + 3_y_ = 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.
Substitute the second equation into the first to find that y = –5. Then plug y = –5 into either equation to find x = 15.
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A store sells 17 coffee mugs for $169. Some of the mugs are $12 each and some are $7 each. How many $7 coffee mugs were sold?
A store sells 17 coffee mugs for $169. Some of the mugs are $12 each and some are $7 each. How many $7 coffee mugs were sold?
The answer is 7.
Write two independent equations that represent the problem.
x + y = 17 and 12_x_ + 7_y_ = 169
If we solve the first equation for x, we get x = 17 – y and we can plug this into the second equation.
12(17 – y) + 7_y_ = 169
204 – 12_y_ + 7_y_ =169
–5_y_ = –35
y = 7
The answer is 7.
Write two independent equations that represent the problem.
x + y = 17 and 12_x_ + 7_y_ = 169
If we solve the first equation for x, we get x = 17 – y and we can plug this into the second equation.
12(17 – y) + 7_y_ = 169
204 – 12_y_ + 7_y_ =169
–5_y_ = –35
y = 7
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