Algebraic Equations - Math
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The square root of the sum of twice a number and five is equal to the square of the difference between that number and two. Which of the following equations could be used to find the number, represented by n?
The square root of the sum of twice a number and five is equal to the square of the difference between that number and two. Which of the following equations could be used to find the number, represented by n?
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We need to translate this word sentence into algebra: The square root of the sum of twice a number and five is equal to the square of the difference between that number and two.
The words "is equal to" denote the equal sign. Everything before it will be to the left of the equal sign, and everything after will be on the right.
The first half of the sentence says "The square root of the sum of twice a number and five." We can represent twice the number as 2n. The sum of twice the number and five can be represented by 2n + 5, because "sum" denotes addition. We need to take the square root of the quantity 2n + 5, which we can write as $\sqrt{2n+5}$. The left side of the equation will be $\sqrt{2n+5}$.
On the right side, we need to find the square of the difference between n and 2. "Difference" tells us to subtract, so we can represent this as n - 2. If we take the square of this quantity, we can represent this as $(n-2)^2$. The right side of the equation is $(n-2)^2$.
Putting both sides together with the equal sign, the entire equation is $\sqrt{2n+5}$ $=(n-2)^2$.
The answer is $\sqrt{2n+5}$ $=(n-2)^2$.
We need to translate this word sentence into algebra: The square root of the sum of twice a number and five is equal to the square of the difference between that number and two.
The words "is equal to" denote the equal sign. Everything before it will be to the left of the equal sign, and everything after will be on the right.
The first half of the sentence says "The square root of the sum of twice a number and five." We can represent twice the number as 2n. The sum of twice the number and five can be represented by 2n + 5, because "sum" denotes addition. We need to take the square root of the quantity 2n + 5, which we can write as $\sqrt{2n+5}$. The left side of the equation will be $\sqrt{2n+5}$.
On the right side, we need to find the square of the difference between n and 2. "Difference" tells us to subtract, so we can represent this as n - 2. If we take the square of this quantity, we can represent this as $(n-2)^2$. The right side of the equation is $(n-2)^2$.
Putting both sides together with the equal sign, the entire equation is $\sqrt{2n+5}$ $=(n-2)^2$.
The answer is $\sqrt{2n+5}$ $=(n-2)^2$.
Solve for X in the following equation: 
Solve for X in the following equation:
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To solve for X, isolate X on one side of the equation. In this case, we are required to subtract one number from X. When we add that number to both sides of the equation, we isolate X and obtain our answer.
To solve for X, isolate X on one side of the equation. In this case, we are required to subtract one number from X. When we add that number to both sides of the equation, we isolate X and obtain our answer.
Which equation can be used to solve the following problem?
Jason is planning a birthday party at a bowling alley. It costs 
to rent the alley and 
per attendee. What is the total cost (
) of the party in terms of 
, where 
equals the total number of attendees?
Which equation can be used to solve the following problem?
Jason is planning a birthday party at a bowling alley. It costs
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The cost per attendee, 
, must be multiplied by the total number of attendees and then added to the base cost, 
, to obtain the total cost.
The cost per attendee,
A family has 7 animals. Write an equation for the number of cats, 
, in terms of the number of dogs, 
. Assume the family has only cats and dogs.
A family has 7 animals. Write an equation for the number of cats,
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In order to sovle for 
, subtract 
from both sides:
In order to sovle for
Sally went shopping and bought two shirts. One shirt cost $15, and she paid a total of $27 for both shirts. What was the cost of the other shirt?
Sally went shopping and bought two shirts. One shirt cost $15, and she paid a total of $27 for both shirts. What was the cost of the other shirt?
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To solve this word problem, set up an algebraic equation, putting 
in for the unknown value of the second shirt. The equation is 
because we know the cost of the two shirts will total 
.
To solve for 
, subtract 
from each side. This results in 
.
To solve this word problem, set up an algebraic equation, putting
To solve for
Rojo Salsa is on sale at a price of 
for 
jars of 
ounces each. Verde Salsa is on sale at a price of 
for 
jars of 
ounces each. Which of the following statements is true?
Rojo Salsa is on sale at a price of
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The statement "A jar of Verde Salsa costs more than a jar of Rojo Salsa" can be tested by comparing the price per jar of each salsa.
versus 
The statement is false since the price of Rojo per jar is greater.
The remaining statements above can all be proven true or false by finding the price per ounce of each salsa.
Rojo Salsa is on sale at a price of 
for 
jars of 
ounces each. The following operations can be used to determine the cost of Rojo Salsa per ounce:
for Rojo Salsa.
Verde Salsa is on sale at a price of 
for 
jars of 
ounces each. The following operations can be used to determine the cost of Verde Salsa per ounce.
for Verde Salsa.
The only true statement is "An ounce of Rojo Salsa is the same price as an ounce of Verde Salsa."
The statement "A jar of Verde Salsa costs more than a jar of Rojo Salsa" can be tested by comparing the price per jar of each salsa.
The statement is false since the price of Rojo per jar is greater.
The remaining statements above can all be proven true or false by finding the price per ounce of each salsa.
Rojo Salsa is on sale at a price of
Verde Salsa is on sale at a price of
The only true statement is "An ounce of Rojo Salsa is the same price as an ounce of Verde Salsa."
Barbara lives 
miles from the beach, and her friend Josef lives 
miles from the beach. If Barbara and Joe leave their homes at the same time, and Barbara drives 
miles per hour, how fast will Joe need to drive to arrive at the beach at the exact same time as Barbara?
Barbara lives
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To find the speed (or rate) that Josef will need to travel, we can use the equation 
(
).
This equation cannot be used for Josef yet, since only the distance traveled is known and not the time in which he will need to make the trip.
To find the time it takes Barbara to make the trip, use the same equation to solve for 
, where the distance is the length of Barbara's trip. Note that we express 
miles per hour as a fraction that represents the ratio of 
miles to 
hour.
Multiply both sides of the equation by the reciprocal of the rate. Note that the unit of "miles" cancels out, leaving only the unit "hours" (time). The result will be expressed as a fraction of a single hour.
The amount of time it takes Barbara to get to the beach must be the same amount of time it takes Joseph to get to the beach.
Therefore, we can use this new value of 
and the 
equation to find the rate Josef will need to travel for his 
mile trip.
Multiply both sides by the reciprocal of time (a fraction) to isolate the rate.
To find the speed (or rate) that Josef will need to travel, we can use the equation
This equation cannot be used for Josef yet, since only the distance traveled is known and not the time in which he will need to make the trip.
To find the time it takes Barbara to make the trip, use the same equation to solve for
Multiply both sides of the equation by the reciprocal of the rate. Note that the unit of "miles" cancels out, leaving only the unit "hours" (time). The result will be expressed as a fraction of a single hour.
The amount of time it takes Barbara to get to the beach must be the same amount of time it takes Joseph to get to the beach.
Therefore, we can use this new value of
Multiply both sides by the reciprocal of time (a fraction) to isolate the rate.
Sarah sells necklaces for 
each. She sells 
necklaces in a month at this price. If she applied a 
% discount to the price of her necklaces, she would sell an additional 
necklaces in a month. How much additional money would Sarah make in sales if she sold her necklaces with the 
% discount for a month?
Sarah sells necklaces for
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To find how much additional money Sarah would make by applying the discount, find the difference between her earnings in a normal month and a month where the discount is applied. In a normal month, multiply the normal price by the normal quantity sold to find the normal earnings:
To find earnings at the discounted price, first calculate how much each necklace will cost with the 
% discount. To do this, subtract the amount discounted (calculated by percent as a fraction of 
multiplied by the original price) from the original price.
If 
necklaces are sold at the new discounted price of 
each, multiply these together to find the total earnings with the discount.
Finally, subtract the earnings without the discount from the earnings with the discount to find the additional money made by applying the discount.
To find how much additional money Sarah would make by applying the discount, find the difference between her earnings in a normal month and a month where the discount is applied. In a normal month, multiply the normal price by the normal quantity sold to find the normal earnings:
To find earnings at the discounted price, first calculate how much each necklace will cost with the
If
Finally, subtract the earnings without the discount from the earnings with the discount to find the additional money made by applying the discount.
A shirt costs $12 after a 15% discount. What was the original price of the shirt?
A shirt costs $12 after a 15% discount. What was the original price of the shirt?
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Convert 15% to a decimal.
Let the original price equal 
. The discount will be 15% of 
. Subtracting the discount from the original price will equal the amount paid, $12.
Using this equation, we can solve for 
.
Convert 15% to a decimal.
Let the original price equal
Using this equation, we can solve for
A farmer has 
units of fence. If he uses this to build a square fence, what will be the length of each side?
A farmer has
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If this is a square fence, then each of the four sides will be equal.
The fence in question will become the perimeter of that square.
Since 
when working with a square, for this problem 
.
If this is a square fence, then each of the four sides will be equal.
The fence in question will become the perimeter of that square.
Since
The area of a rectangle is 
. How many whole 
by 
rectangles can fit inside of this larger rectangle?
The area of a rectangle is
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First we need to find the area of the smaller rectangle.
Now to find out how many can fit, we divide the total area by the smaller area.
However, the problem is asking how many WHOLE rectangles can fit. Therefore only 
can fit.
First we need to find the area of the smaller rectangle.
Now to find out how many can fit, we divide the total area by the smaller area.
However, the problem is asking how many WHOLE rectangles can fit. Therefore only
40% of Eric's socks are black, and the rest are white. If he has 6 pairs of black socks, how many pairs of white socks does he have?
40% of Eric's socks are black, and the rest are white. If he has 6 pairs of black socks, how many pairs of white socks does he have?
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First, use the information to find the total number of socks Eric has by setting up a proportion:
, where 
is the number of total socks.
Once you cross multiply and divide, you will find that 
.
To find the number of white socks Eric has, subtract the number of black socks 
from the total number of socks:
First, use the information to find the total number of socks Eric has by setting up a proportion:
Once you cross multiply and divide, you will find that
To find the number of white socks Eric has, subtract the number of black socks
A chemistry student has 
of a 
acid solution. She needs a 
acid solution for an experiement. How much pure water should she add?
A chemistry student has
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pure water and 
pure acid
In general, mixture problems have the form:
, where 
volume and 
percent
The equation to solve becomes:
Then the solution is 
.
In general, mixture problems have the form:
The equation to solve becomes:
Then the solution is
Sarah is having a birthday party. She wants to invite 
friends. Each invitation costs 
, and each stamp costs 
. If she plans to mail all of the invitations, how much change should she get back if she pays with a 
bill?
Sarah is having a birthday party. She wants to invite
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The cost of mailing each invitation is 
.
To invite everyone, it will cost 
.
Then the change is 
.
The cost of mailing each invitation is
To invite everyone, it will cost
Then the change is
What number is 
of 
?
What number is
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Look for the verbal cues in the question. IS translates to "equals" and OF translates to multiplication.
Thus the equation to solve becomes:
Look for the verbal cues in the question. IS translates to "equals" and OF translates to multiplication.
Thus the equation to solve becomes:
What percent of 
is 
?
What percent of
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Look for the verbal cues in the question. IS represents "equals" and OF represents multiplication.
Then the equation to solve becomes:
so 
or 
Look for the verbal cues in the question. IS represents "equals" and OF represents multiplication.
Then the equation to solve becomes:
Joy earns a 
commission on each house she sells. How much money does she make when she sells two houses for 
each and another house for 
?
Joy earns a
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In general, a commission problem has the form:
So the commission is:
In general, a commission problem has the form:
So the commission is:
Dean is counting his money. He has five more quarters than nickles and four less nickles than dimes. He has 
. How many coins does he have?
Dean is counting his money. He has five more quarters than nickles and four less nickles than dimes. He has
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Money problems generally consist of two parts: total number of coins and total value of money.
Let 
number of nickles, 
number of quarters, and 
number of dimes.
So the equation to solve becomes:
or
so
Thus, 
, so there are three nickles, eight quarters and seven dimes for a total of 
coins.
Money problems generally consist of two parts: total number of coins and total value of money.
Let
So the equation to solve becomes:
Thus,
Edward is making cookies. The full recipe makes 
cookies. He decides to cut the recipe by four. How many dozens of cookies will he make?
Edward is making cookies. The full recipe makes
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Cut the recipe by four: 
cookies; however, the question asks for how many DOZENS of cookies are made, so the correct answer is 
dozen cookies.
Cut the recipe by four:
Golden Sunshine paint is made by mixing one part red paint and three parts yellow paint. How many gallons of yellow paint should be mixed with three quarts of red paint?
Golden Sunshine paint is made by mixing one part red paint and three parts yellow paint. How many gallons of yellow paint should be mixed with three quarts of red paint?
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Use proportions to solve this problem.
Let 
quarts of yellow paint.
By cross-multiplying we get 
.
Since 
, we actually need 
.
Use proportions to solve this problem.
Let
By cross-multiplying we get
Since