How to find the solution to an equation - ACT Math
Card 0 of 1134
11/(x – 7) + 4/(7 – x) = ?
11/(x – 7) + 4/(7 – x) = ?
We must find a common denominator and here they changed the first fraction by removing a negative from the numerator and denominator, leaving –11/(7 – x). We add the numerators and keep the same denominator to find the answer.
We must find a common denominator and here they changed the first fraction by removing a negative from the numerator and denominator, leaving –11/(7 – x). We add the numerators and keep the same denominator to find the answer.
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The population of a bird species is modeled by the following equation:
,
where 
represents the number of years from the present. How many years will it take the population to reach 130 birds (rounded to the nearest tenth)?
The population of a bird species is modeled by the following equation:
where
Plugging in 130 for P, the equation becomes 130 = (11/8)x + 102. Solving for x, we obtain x = 20.4 years.
Plugging in 130 for P, the equation becomes 130 = (11/8)x + 102. Solving for x, we obtain x = 20.4 years.
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If bx + c = e – ax, then what is x?
If bx + c = e – ax, then what is x?
To solve for x:
bx + c = e – ax
bx + ax = e – c
x(b+a) = e-c
x = (e-c) / (b+a)
To solve for x:
bx + c = e – ax
bx + ax = e – c
x(b+a) = e-c
x = (e-c) / (b+a)
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A sequence of numbers is: 2, 5, 8, 11. Assuming it follows the same pattern, what would be the value of the 20th number?
A sequence of numbers is: 2, 5, 8, 11. Assuming it follows the same pattern, what would be the value of the 20th number?
This goes up at a constant number between values, making it an arthmetic sequence. The first number is 2, with a difference of 3. Plugging this into the arithmetic equation you get An = 2 + 3 (n – 1). Plugging in 20 for n, you get a value of 59.
This goes up at a constant number between values, making it an arthmetic sequence. The first number is 2, with a difference of 3. Plugging this into the arithmetic equation you get An = 2 + 3 (n – 1). Plugging in 20 for n, you get a value of 59.
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The first four numbers of a sequence are 5, 10, 20, 40. Assuming the pattern continues, what is the 6th term of the sequence?
The first four numbers of a sequence are 5, 10, 20, 40. Assuming the pattern continues, what is the 6th term of the sequence?
Looking at the sequence you can see that it doubles each term, making it a geometric sequence. Since it doubles r = 2 and the first term is 5. Plugging this into the geometric equation you get An = 5(2)n–1. Setting n = 6, you get 160 as the 6th term.
Looking at the sequence you can see that it doubles each term, making it a geometric sequence. Since it doubles r = 2 and the first term is 5. Plugging this into the geometric equation you get An = 5(2)n–1. Setting n = 6, you get 160 as the 6th term.
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Given f(x) = x2 – 9. What are the zeroes of the function?
Given f(x) = x2 – 9. What are the zeroes of the function?
The zeroes of the equation are where f(x) = 0 (aka x-intercepts). Setting the equation equal to zero you get x2 = 9. Since a square makes a negative number positive, x can be equal to 3 or –3.
The zeroes of the equation are where f(x) = 0 (aka x-intercepts). Setting the equation equal to zero you get x2 = 9. Since a square makes a negative number positive, x can be equal to 3 or –3.
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Give the lines y = 0.5x+3 and y=3x-2. What is the y value of the point of intersection?
Give the lines y = 0.5x+3 and y=3x-2. What is the y value of the point of intersection?
In order to solve for the x value you set both equations equal to each other (0.5x+3=3x-2). This gives you the x value for the point of intersection at x=2. Plugging x=2 into either equation gives you y=4.
In order to solve for the x value you set both equations equal to each other (0.5x+3=3x-2). This gives you the x value for the point of intersection at x=2. Plugging x=2 into either equation gives you y=4.
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If 12x + 3 = 2(5x + 5) + 1, what is the value of x?
If 12x + 3 = 2(5x + 5) + 1, what is the value of x?
Starting with 12x + 3 = 2(5x + 5) + 1, we start by solving the parenthesis, giving us 12x + 3 = 10x + 11. We then subtract 10x from the right side and subtract three from the left, giving us 2x = 8; divide by 2 → x = 4.
Starting with 12x + 3 = 2(5x + 5) + 1, we start by solving the parenthesis, giving us 12x + 3 = 10x + 11. We then subtract 10x from the right side and subtract three from the left, giving us 2x = 8; divide by 2 → x = 4.
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If you multiply two integers together and then add 5, the result is 69. Which of the following could not be the sum of the two integers?
If you multiply two integers together and then add 5, the result is 69. Which of the following could not be the sum of the two integers?
The equation is xy + 5 = 69, making xy equal to 64. If we factor 64, we see that 1x64, 2x32, 4x16 and 8x8 all equal 16 when the two numbers are added together, so 24 is the only possibility that does not work.
The equation is xy + 5 = 69, making xy equal to 64. If we factor 64, we see that 1x64, 2x32, 4x16 and 8x8 all equal 16 when the two numbers are added together, so 24 is the only possibility that does not work.
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If 5 + x is 5 more than 5,what is the value of 2_x_?
If 5 + x is 5 more than 5,what is the value of 2_x_?
5 more than 5 = 10
5 + x = 10
Subtract 5 from each side of the equation: x = 5 → 2_x_ = 10
5 more than 5 = 10
5 + x = 10
Subtract 5 from each side of the equation: x = 5 → 2_x_ = 10
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Three consecutive positive numbers have the sum of 15. What is the product of these numbers?
Three consecutive positive numbers have the sum of 15. What is the product of these numbers?
Define the variables as x = the first number, x + 1, the second number, and x + 2 the thrid number.
The sum becomes x + x + 1 + x + 2 = 15 so 3x + 3 = 15. Subtract 3 from both sides of the equation to get 3x = 12 → 3x/3 = 12/3 → x = 4
The three numbers are 4, 5, and 6 and their product is 120.
Define the variables as x = the first number, x + 1, the second number, and x + 2 the thrid number.
The sum becomes x + x + 1 + x + 2 = 15 so 3x + 3 = 15. Subtract 3 from both sides of the equation to get 3x = 12 → 3x/3 = 12/3 → x = 4
The three numbers are 4, 5, and 6 and their product is 120.
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A company makes toy boats. Their monthly fixed costs are $1500. The variable costs are $50 per boat. They sell boats for $75 a piece. How many boats must be sold each month to break even?
A company makes toy boats. Their monthly fixed costs are $1500. The variable costs are $50 per boat. They sell boats for $75 a piece. How many boats must be sold each month to break even?
The break-even point is where the costs equal the revenues
Fixed Costs + Variable Costs = Revenues
1500 + 50_x_ = 75_x_
Solving for x results in x = 60 boats sold each month to break even.
The break-even point is where the costs equal the revenues
Fixed Costs + Variable Costs = Revenues
1500 + 50_x_ = 75_x_
Solving for x results in x = 60 boats sold each month to break even.
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Jack has 14 coins consisting of nickels and dimes that total $0.90. How many nickels does Jack have?
Jack has 14 coins consisting of nickels and dimes that total $0.90. How many nickels does Jack have?
In order to solve this question we must first set up two equations. We know the number of nickels and the number of dimes equals 14 (n + d = 14). We also know the value of nickels and dimes.
For the second equation we simply multiply the number of nickels we have by their value, added to the number of dimes we have by their value to get the total (0.05n + 0.10d = 0.90).
Solve the first equation for n giving us n = 14 – d. We can then substitute 14 – d into the second equation wherever there is an “n”. Giving us 0.05 (14 – d) + 0.10d = 0.90.
When we solve the equation we find the number of dimes is d = 4; therefore the remaining 10 coins must be nickels.
In order to solve this question we must first set up two equations. We know the number of nickels and the number of dimes equals 14 (n + d = 14). We also know the value of nickels and dimes.
For the second equation we simply multiply the number of nickels we have by their value, added to the number of dimes we have by their value to get the total (0.05n + 0.10d = 0.90).
Solve the first equation for n giving us n = 14 – d. We can then substitute 14 – d into the second equation wherever there is an “n”. Giving us 0.05 (14 – d) + 0.10d = 0.90.
When we solve the equation we find the number of dimes is d = 4; therefore the remaining 10 coins must be nickels.
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If a = 1/3b and b = 4c, then in terms of c, a – b + c = ?
If a = 1/3b and b = 4c, then in terms of c, a – b + c = ?
To begin we must find how a and c relate to each other. Using the second equation we know that we can plug in 4c everywhere there is a b in the first equation, giving us a = 4/3c.
Now we can plug into the last equation. We plug in 4/3c for a, 4c for b, and leave c as it is. We must find a common denominator (4/3c – 12/3c + 3/3c) and add the numerators to find that our equation equals –5/3c.
To begin we must find how a and c relate to each other. Using the second equation we know that we can plug in 4c everywhere there is a b in the first equation, giving us a = 4/3c.
Now we can plug into the last equation. We plug in 4/3c for a, 4c for b, and leave c as it is. We must find a common denominator (4/3c – 12/3c + 3/3c) and add the numerators to find that our equation equals –5/3c.
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If x3 = 8, then x2(4/(3 – x))(2/(4 – x)) – (4/x2) = ?
If x3 = 8, then x2(4/(3 – x))(2/(4 – x)) – (4/x2) = ?
There is really no need to alter this equation using algebra. Simply find that x = 2 and plug in. We see that 4(4)(1) – (1)=15. Remember to use correct order of operations here (parentheses, exponents, multiplication, division, addition, subtraction).
There is really no need to alter this equation using algebra. Simply find that x = 2 and plug in. We see that 4(4)(1) – (1)=15. Remember to use correct order of operations here (parentheses, exponents, multiplication, division, addition, subtraction).
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If A + B = C and B – C = 3, what does A equal?
If A + B = C and B – C = 3, what does A equal?
This question can be solved using substitution. There are several ways to subsitute, but one possible method is to substitue (A+B) for C in the equation B-C=3: B-(A+B)=3. This simplifies to (B-A-B=3) (don't forget to distribute the negative!). Since (B-B)=0, -A must equal 3 (-A=3). If we divide both sides by -1, A=-3. If you got an answer of positive 3, you may have forgotten to divide both sides by -1.
This question can be solved using substitution. There are several ways to subsitute, but one possible method is to substitue (A+B) for C in the equation B-C=3: B-(A+B)=3. This simplifies to (B-A-B=3) (don't forget to distribute the negative!). Since (B-B)=0, -A must equal 3 (-A=3). If we divide both sides by -1, A=-3. If you got an answer of positive 3, you may have forgotten to divide both sides by -1.
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If y=0.35(300 – 2y), what is the value of y to the nearest tenth?
If y=0.35(300 – 2y), what is the value of y to the nearest tenth?
Distributing the 0.35 to both the 300 and **–**2y leaves y=105 – 0.7y
Adding 0.7y to both sides and dividing by 1.7 gives 61.8.
Distributing the 0.35 to both the 300 and **–**2y leaves y=105 – 0.7y
Adding 0.7y to both sides and dividing by 1.7 gives 61.8.
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Find the intersection of the following two equations:
3x + 4y = 6
15x - 4y = 3
Find the intersection of the following two equations:
3x + 4y = 6
15x - 4y = 3
The point of intersection for two lines is the same as the values of x and y that mutually solve each equation. Although you could solve for one variable and replace it in the other equation, use elementary row operations to add the two equations since you have a 4y and -4y:
3x + 4y = 6
15x - 4y = 3
18x = 9; x = 0.5
You can now plug x into the first equation:
3 * 0.5 + 4y = 6; 1.5 +4y = 6; 4y = 4.5; y = 1.125
Therefore, our point of intersection is (0.5, 1.125)
The point of intersection for two lines is the same as the values of x and y that mutually solve each equation. Although you could solve for one variable and replace it in the other equation, use elementary row operations to add the two equations since you have a 4y and -4y:
3x + 4y = 6
15x - 4y = 3
18x = 9; x = 0.5
You can now plug x into the first equation:
3 * 0.5 + 4y = 6; 1.5 +4y = 6; 4y = 4.5; y = 1.125
Therefore, our point of intersection is (0.5, 1.125)
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A store sells potatoes for $0.24 and tomatoes for $0.76. Fred bought 12 individual vegetables. If he paid $6.52 total, how many potatoes did Fred buy?
A store sells potatoes for $0.24 and tomatoes for $0.76. Fred bought 12 individual vegetables. If he paid $6.52 total, how many potatoes did Fred buy?
Set up an equation to represent the total cost in cents: 24P + 76T = 652. In order to reduce the number of variables from 2 to 1, let the # tomatoes = 12 – # of potatoes. This makes the equation 24P + 76(12 – P) = 652.
Solving for P will give the answer.
Set up an equation to represent the total cost in cents: 24P + 76T = 652. In order to reduce the number of variables from 2 to 1, let the # tomatoes = 12 – # of potatoes. This makes the equation 24P + 76(12 – P) = 652.
Solving for P will give the answer.
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Kim is twice as old as Claire. Nick is 3 years older than Claire. Kim is 6 years older than Emily. Their ages combined equal 81. How old is Nick?
Kim is twice as old as Claire. Nick is 3 years older than Claire. Kim is 6 years older than Emily. Their ages combined equal 81. How old is Nick?
The goal in this problem is to have only one variable. Variable “x” can designate Claire’s age.
Then Nick is x + 3, Kim is 2x, and Emily is 2x – 6; therefore x + x + 3 + 2x + 2x – 6 = 81
Solving for x gives Claire’s age, which can be used to find Nick’s age.
The goal in this problem is to have only one variable. Variable “x” can designate Claire’s age.
Then Nick is x + 3, Kim is 2x, and Emily is 2x – 6; therefore x + x + 3 + 2x + 2x – 6 = 81
Solving for x gives Claire’s age, which can be used to find Nick’s age.
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