Solving Linear Equations in Word Problems - SAT Math
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Erin is making thirty shirts for her upcoming family reunion. At the reunion she is selling each shirt for $18 apiece. If each shirt cost her $10 apiece to make, how much profit does she make if she only sells 25 shirts at the reunion?
Erin is making thirty shirts for her upcoming family reunion. At the reunion she is selling each shirt for $18 apiece. If each shirt cost her $10 apiece to make, how much profit does she make if she only sells 25 shirts at the reunion?
This problem involves two seperate multiplication problems. Erin will make $450 at the reunion but supplies cost her $300 to make the shirts. So her profit is $150.
This problem involves two seperate multiplication problems. Erin will make $450 at the reunion but supplies cost her $300 to make the shirts. So her profit is $150.
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Write as an equation:
"Ten added to the product of a number and three is equal to twice the number."
Write as an equation:
"Ten added to the product of a number and three is equal to twice the number."
Let 
represent the unknown quantity.
The first expression:
"The product of a number and three" is three times this number, or
"Ten added to the product" is
The second expression:
"Twice the number" is two times the number, or
.
The desired equation is therefore
.
Let
The first expression:
"The product of a number and three" is three times this number, or
"Ten added to the product" is
The second expression:
"Twice the number" is two times the number, or
The desired equation is therefore
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Write as an equation:
Five-sevenths of the difference of a number and nine is equal to forty.
Write as an equation:
Five-sevenths of the difference of a number and nine is equal to forty.
"The difference of a number and nine" is the result of a subtraction of the two, so we write this as
"Five-sevenths of" this difference is the product of 
and this, or
This is equal to forty, so write the equation as
"The difference of a number and nine" is the result of a subtraction of the two, so we write this as
"Five-sevenths of" this difference is the product of
This is equal to forty, so write the equation as
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Write as an equation:
Twice the sum of a number and ten is equal to the difference of the number and one half.
Write as an equation:
Twice the sum of a number and ten is equal to the difference of the number and one half.
Let 
represent the unknown number.
"The sum of a number and ten" is the expression 
. "Twice" this sum is two times this expression, or
.
"The difference of the number and one half" is a subtraction of the two, or
Set these equal, and the desired equation is
Let
"The sum of a number and ten" is the expression
"The difference of the number and one half" is a subtraction of the two, or
Set these equal, and the desired equation is
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If a rectangle possesses a width of 
and has a perimeter of 
, then what is the length?
If a rectangle possesses a width of
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the 
to one side of the equation. In order to do this, we will first subtract 
from both sides of the equation.
Next, we can divide each side by 
The length of the rectangle is 
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the
Next, we can divide each side by
The length of the rectangle is
Compare your answer with the correct one above
If a rectangle possesses a width of 
and has a perimeter of 
, then what is the length?
If a rectangle possesses a width of
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the to one side of the equation. In order to do this, we will first subtract 
from both sides of the equation.
Next, we can divide each side by
The length of the rectangle is 
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the
Next, we can divide each side by
The length of the rectangle is
Compare your answer with the correct one above
If a rectangle possesses a width of 
and has a perimeter of 
, then what is the length?
If a rectangle possesses a width of
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the to one side of the equation. In order to do this, we will first subtract 
from both sides of the equation.
Next, we can divide each side by
The length of the rectangle is 
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the
Next, we can divide each side by
The length of the rectangle is
Compare your answer with the correct one above
If a rectangle possesses a width of 
and has a perimeter of 
, then what is the length?
If a rectangle possesses a width of
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the to one side of the equation. In order to do this, we will first subtract 
from both sides of the equation.
Next, we can divide each side by
The length of the rectangle is 
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the
Next, we can divide each side by
The length of the rectangle is
Compare your answer with the correct one above
If a rectangle possesses a width of 
and has a perimeter of 
, then what is the length?
If a rectangle possesses a width of
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the to one side of the equation. In order to do this, we will first subtract 
from both sides of the equation.
Next, we can divide each side by
The length of the rectangle is 
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the
Next, we can divide each side by
The length of the rectangle is
Compare your answer with the correct one above
If a rectangle possesses a width of 
and has a perimeter of 
, then what is the length?
If a rectangle possesses a width of
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the to one side of the equation. In order to do this, we will first subtract 
from both sides of the equation.
Next, we can divide each side by
The length of the rectangle is 
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the
Next, we can divide each side by
The length of the rectangle is
Compare your answer with the correct one above
If a rectangle possesses a width of 
and has a perimeter of 
, then what is the length?
If a rectangle possesses a width of
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the to one side of the equation. In order to do this, we will first subtract 
from both sides of the equation.
Next, we can divide each side by
The length of the rectangle is 
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the
Next, we can divide each side by
The length of the rectangle is
Compare your answer with the correct one above
If a rectangle possesses a width of 
and has a perimeter of 
, then what is the length?
If a rectangle possesses a width of
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the to one side of the equation. In order to do this, we will first subtract 
from both sides of the equation.
Next, we can divide each side by
The length of the rectangle is 
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the
Next, we can divide each side by
The length of the rectangle is
Compare your answer with the correct one above
If a rectangle possesses a width of 
and has a perimeter of 
, then what is the length?
If a rectangle possesses a width of
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the to one side of the equation. In order to do this, we will first subtract 
from both sides of the equation.
Next, we can divide each side by
The length of the rectangle is 
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the
Next, we can divide each side by
The length of the rectangle is
Compare your answer with the correct one above
If a rectangle possesses a width of 
and has a perimeter of 
, then what is the length?
If a rectangle possesses a width of
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the to one side of the equation. In order to do this, we will first subtract 
from both sides of the equation.
Next, we can divide each side by
The length of the rectangle is 
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the
Next, we can divide each side by
The length of the rectangle is
Compare your answer with the correct one above
If a rectangle possesses a width of 
and has a perimeter of 
, then what is the length?
If a rectangle possesses a width of
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the to one side of the equation. In order to do this, we will first subtract 
from both sides of the equation.
Next, we can divide each side by
The length of the rectangle is 
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the
Next, we can divide each side by
The length of the rectangle is
Compare your answer with the correct one above
If a rectangle possesses a width of 
and has a perimeter of 
, then what is the length?
If a rectangle possesses a width of
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the to one side of the equation. In order to do this, we will first subtract 
from both sides of the equation.
Next, we can divide each side by
The length of the rectangle is 
In order to solve this problem, we need to recall the formula for perimeter of a rectangle:
We can substitute in our known values and solve for our unknown variable (i.e. length):
We want to isolate the
Next, we can divide each side by
The length of the rectangle is
Compare your answer with the correct one above
Jackie plans to buy one video game and a number of hardcover books. The video game costs $40 and each book costs $30. If she must spend at least $120 in order to get free shipping, what is the minimum number of books she must buy in order to get free shipping?
Jackie plans to buy one video game and a number of hardcover books. The video game costs $40 and each book costs $30. If she must spend at least $120 in order to get free shipping, what is the minimum number of books she must buy in order to get free shipping?
Since you know that Jackie will purchase exactly one video game for $40, you can set up your equation (or inequality) here as:
Where 
represents the number of books she buys. Since we're looking for the minimum number of books she needs to buy in order to be greater than or equal to $120, we use the greater-than-or-equal-to inequality.
Now you can subtract 40 from both sides:
And when you divide both sides by 30 you'll see that you have a number between 2 and 3. Note that you do not have to do the full decimal calculation, because you cannot buy part of a book! So since she can't buy 2.67 books, the minimum number she can purchase is 3.
Since you know that Jackie will purchase exactly one video game for $40, you can set up your equation (or inequality) here as:
Where 
Now you can subtract 40 from both sides:
And when you divide both sides by 30 you'll see that you have a number between 2 and 3. Note that you do not have to do the full decimal calculation, because you cannot buy part of a book! So since she can't buy 2.67 books, the minimum number she can purchase is 3.
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On the first day of the week, a bakery had an inventory of 450 loaves of bread. It bakes 210 loaves of bread and sells 240 loaves of bread each day that it is open, and then closes for a baking day when it runs out of loaves. How many days can it be open before it must close for a baking day?
On the first day of the week, a bakery had an inventory of 450 loaves of bread. It bakes 210 loaves of bread and sells 240 loaves of bread each day that it is open, and then closes for a baking day when it runs out of loaves. How many days can it be open before it must close for a baking day?
If the bakery bakes 210 loaves of bread and sells 240 loaves of bread, then that means that, total, it loses 30 loaves of bread per day. Since you know that it starts with 450 loaves of bread, you can use this information to write a linear equation relating the number of loaves of bread left with how many days it has been since the bakery has closed for a baking day. It should look like:
B = 450 - 30d
Where B represents the number of loaves left and d represents the number of days since the bakery’s last baking day.
In order for the bakery to need to close for a baking day, the number of loaves left must equal 0. If you substitute in B = 0, you get: 0 = 450−30d.
Now you can solve: add 30d to both sides to get:
30d = 450
And then divide both sides by 30 to get d = 15
If the bakery bakes 210 loaves of bread and sells 240 loaves of bread, then that means that, total, it loses 30 loaves of bread per day. Since you know that it starts with 450 loaves of bread, you can use this information to write a linear equation relating the number of loaves of bread left with how many days it has been since the bakery has closed for a baking day. It should look like:
B = 450 - 30d
Where B represents the number of loaves left and d represents the number of days since the bakery’s last baking day.
In order for the bakery to need to close for a baking day, the number of loaves left must equal 0. If you substitute in B = 0, you get: 0 = 450−30d.
Now you can solve: add 30d to both sides to get:
30d = 450
And then divide both sides by 30 to get d = 15
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Katharine currently has $10,000 saved for a down payment on purchasing her first house, which she can do when her savings has reached $50,000. Each month she earns $6,500 but incurs $4,000 in expenses. If her earnings and expenses remain constant, how many months will it take until she has reached her savings goal?
Katharine currently has $10,000 saved for a down payment on purchasing her first house, which she can do when her savings has reached $50,000. Each month she earns $6,500 but incurs $4,000 in expenses. If her earnings and expenses remain constant, how many months will it take until she has reached her savings goal?
To turn this problem into an equation, you can start by putting Katharine's goal of $50,000 on one side of the equation, and then arranging all of the inputs to that goal (ways she earns money) and impediments (ways she loses money) on the other. If you use m to represent the number of months, an initial equation would look like:
$50,000 = $10,000 + $6,500m - $4,000m
Which, qualitatively, is that her goal of $50,000 is equal to the $10,000 she has plus the $6,500 she earns each month, minus the $4,000 she loses each month.
Now you can combine like terms to simplify the equation:
$40,000 = $2,500m
And then divide 40000 by 2500 (which simplifies to 400 divided by 25) to solve:
m = 16 months
To turn this problem into an equation, you can start by putting Katharine's goal of $50,000 on one side of the equation, and then arranging all of the inputs to that goal (ways she earns money) and impediments (ways she loses money) on the other. If you use m to represent the number of months, an initial equation would look like:
$50,000 = $10,000 + $6,500m - $4,000m
Which, qualitatively, is that her goal of $50,000 is equal to the $10,000 she has plus the $6,500 she earns each month, minus the $4,000 she loses each month.
Now you can combine like terms to simplify the equation:
$40,000 = $2,500m
And then divide 40000 by 2500 (which simplifies to 400 divided by 25) to solve:
m = 16 months
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If 75 gallons of water were added to a pool that is half full, the pool would then be 
full. How many gallons of water does the pool hold when it is full?
If 75 gallons of water were added to a pool that is half full, the pool would then be 
To translate this word problem into equation form, note that your unknown is the capacity of the pool. You know that the pool is currently half full and that with the proposed addition it would be 2/3 full, but you need to assign those fractions to an actual value. So your variable, 
, will represent the total capacity of the pool. Then you can set up your problem as:
From here it's an algebra problem. To get your 
terms together, subtract 
from both sides:
Then you'll need to find a common denominator of 6, and rewrite what you have to reflect that common denominator:
You can then combine like terms on the right-hand side of the equation:
And then multiply both sides by 6 to finish the problem:
To translate this word problem into equation form, note that your unknown is the capacity of the pool. You know that the pool is currently half full and that with the proposed addition it would be 2/3 full, but you need to assign those fractions to an actual value. So your variable, 
From here it's an algebra problem. To get your 
Then you'll need to find a common denominator of 6, and rewrite what you have to reflect that common denominator:
You can then combine like terms on the right-hand side of the equation:
And then multiply both sides by 6 to finish the problem:
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