Basic Single-Variable Algebra - Algebra 2
Card 1 of 3288
Emeka is a professional fitness instructor who will be teaching classes at a local gym. To get certified as an instructor, he spent a total of $500. Emeka will be earning a base salary of $100 per month from the gym, plus an additional $20 for every class he teaches. If Emeka teaches a certain number of classes during his first month as an instructor, he will earn back the amount he spent on certification. How many classes will that take?
Write a system of equations that models this situation, graph it, and type the solution.
Emeka is a professional fitness instructor who will be teaching classes at a local gym. To get certified as an instructor, he spent a total of $500. Emeka will be earning a base salary of $100 per month from the gym, plus an additional $20 for every class he teaches. If Emeka teaches a certain number of classes during his first month as an instructor, he will earn back the amount he spent on certification. How many classes will that take?
Write a system of equations that models this situation, graph it, and type the solution.
Tap to reveal answer
To start, we want to write two equations that model this situation. Those equations are:
(where c is the number of classes Emeka teaches)
The graph of these two equations is below:
Finally, the intersection of these two lines is associated with the total number of classes Emeka needs to teach to break even. We can see that the intersection is at (20, 500). Therefore, after teaching 20 classes, Emeka will have made $500 from the gym and broke even on his investment.
To start, we want to write two equations that model this situation. Those equations are:
The graph of these two equations is below:
Finally, the intersection of these two lines is associated with the total number of classes Emeka needs to teach to break even. We can see that the intersection is at (20, 500). Therefore, after teaching 20 classes, Emeka will have made $500 from the gym and broke even on his investment.
← Didn't Know|Knew It →
This graph shows a solution at its intersection, the point (1,-1). This solution could be the solution for which of the following systems of equations?
This graph shows a solution at its intersection, the point (1,-1). This solution could be the solution for which of the following systems of equations?
Tap to reveal answer
Based on the graph, you can see that the purple line has the equation 
based on the fact that it goes through the points (0,0) and (1,-1). The red line goes through the points (0,-4) and (1,-1). We can see that its slope is 
and its y-intercept is at -4. Putting this into y=mx+b slope intercept form, we get 
. Therefore, this graph is the solution to the system of equations
Based on the graph, you can see that the purple line has the equation 
← Didn't Know|Knew It →
Alf and Tanner are having a workout competition. Whoever completes the most workouts in a month gets $25 from the other. Alf has already completed 4 workouts, whereas Tanner has finished 2 of them. Going forward, Alf has committed to completing 2 workouts per week, and Tanner has committed to completing 3 per week. At some point soon, the two friends will have completed the same number of workouts. How many workouts will that be?
Write a system of equations, graph them, and type the solution.
Alf and Tanner are having a workout competition. Whoever completes the most workouts in a month gets $25 from the other. Alf has already completed 4 workouts, whereas Tanner has finished 2 of them. Going forward, Alf has committed to completing 2 workouts per week, and Tanner has committed to completing 3 per week. At some point soon, the two friends will have completed the same number of workouts. How many workouts will that be?
Write a system of equations, graph them, and type the solution.
Tap to reveal answer
A system of equations that models this, where a is the number of workouts Alf has completed and t is the number of workouts Tanner has completed is:
Next, graph each of these equations:
Then, identify where the two lines meet each other. This is at (2,8). Therefore, after they each complete 8 workouts (in 2 weeks time) they will have completed the same number of workouts.
A system of equations that models this, where a is the number of workouts Alf has completed and t is the number of workouts Tanner has completed is:
Next, graph each of these equations:
Then, identify where the two lines meet each other. This is at (2,8). Therefore, after they each complete 8 workouts (in 2 weeks time) they will have completed the same number of workouts.
← Didn't Know|Knew It →
This graph shows which of the following inequalities?
This graph shows which of the following inequalities?
Tap to reveal answer
The function graphed is 
. We know that it is strictly less than because the line is dotted (not solid). We also know that y is less than because the portion of the graph below the function is shaded.
The function graphed is 
← Didn't Know|Knew It →
Miles is a professional fitness instructor who will be teaching classes at a local gym. To get certified as an instructor, he spent a total of $400. Miles will be earning a base salary of $80 per month from the gym, plus an additional $20 for every class he teaches. If Miles teaches a certain number of classes during his first month as an instructor, he will earn back the amount he spent on certification. How many classes will that take?
Write a system of equations that models this situation, graph it, and type the solution.
Miles is a professional fitness instructor who will be teaching classes at a local gym. To get certified as an instructor, he spent a total of $400. Miles will be earning a base salary of $80 per month from the gym, plus an additional $20 for every class he teaches. If Miles teaches a certain number of classes during his first month as an instructor, he will earn back the amount he spent on certification. How many classes will that take?
Write a system of equations that models this situation, graph it, and type the solution.
Tap to reveal answer
To start, we want to write two equations that model this situation. Those equations are:
(where c is the number of classes Miles teaches)
The graph of these two equations is below:
Finally, the intersection of these two lines is associated with the total number of classes Miles needs to teach to break even. We can see that the intersection is at (16, 400). Therefore, after teaching 16 classes, Miles will have made $400 from the gym and broke even on his investment.
To start, we want to write two equations that model this situation. Those equations are:
The graph of these two equations is below:
Finally, the intersection of these two lines is associated with the total number of classes Miles needs to teach to break even. We can see that the intersection is at (16, 400). Therefore, after teaching 16 classes, Miles will have made $400 from the gym and broke even on his investment.
← Didn't Know|Knew It →
When graphed, the lines represented in the system of equations below will intersect in which quadrant?
3x - 2y = 0
4x+3y = -17
When graphed, the lines represented in the system of equations below will intersect in which quadrant?
3x - 2y = 0
4x+3y = -17
Tap to reveal answer
Recognize that the intersection of two lines occurs when the two lines meet at the same point - that is, where their x- and y-coordinates are the same. So you can find the point of intersection by simply solving the system. Here that is likely best done through the Elimination Method. If you stack the equations:
3x - 2y = 0
4x+3y = -17
You should see that you have a negative y term in the first equation and a positive y term in the second. That means that if you can get the coefficients of y to be the same, you can sum the equations and eliminate the y-term altogether. To do that, multiply the top equation by 3 and the bottom equation by 2 so that your y-coefficients are each 6:
3(3x - 2y = 0) --> 9x - 6y = 0
2(4x + 3y = -17) --> 8x + 6y = -34
When you sum the equations, then, you arrive at:
17x = -34
Which means that x = -2.
Then plug x = -2 into one of the equations. If you use the first, you'll get:
3(-2) - 2y = 0
Meaning that:
-6 - 2y = 0
-6=2y
y = -3
Since the point of intersection is (-2, -3), with both coordinates negative, the point will lie in Quadrant III.
Recognize that the intersection of two lines occurs when the two lines meet at the same point - that is, where their x- and y-coordinates are the same. So you can find the point of intersection by simply solving the system. Here that is likely best done through the Elimination Method. If you stack the equations:
3x - 2y = 0
4x+3y = -17
You should see that you have a negative y term in the first equation and a positive y term in the second. That means that if you can get the coefficients of y to be the same, you can sum the equations and eliminate the y-term altogether. To do that, multiply the top equation by 3 and the bottom equation by 2 so that your y-coefficients are each 6:
3(3x - 2y = 0) --> 9x - 6y = 0
2(4x + 3y = -17) --> 8x + 6y = -34
When you sum the equations, then, you arrive at:
17x = -34
Which means that x = -2.
Then plug x = -2 into one of the equations. If you use the first, you'll get:
3(-2) - 2y = 0
Meaning that:
-6 - 2y = 0
-6=2y
y = -3
Since the point of intersection is (-2, -3), with both coordinates negative, the point will lie in Quadrant III.
← Didn't Know|Knew It →
The monthly cost to insure your cars varies directly with the number of cars you own. Right now, you are paying $420 per month to insure 3 cars, but you plan to get 2 more cars, so that you will own 5 cars. How much does it cost to insure 5 cars monthly?
The monthly cost to insure your cars varies directly with the number of cars you own. Right now, you are paying $420 per month to insure 3 cars, but you plan to get 2 more cars, so that you will own 5 cars. How much does it cost to insure 5 cars monthly?
Tap to reveal answer
The statement, 'The monthly costly to insure your cars varies directly with the number of cars you own' can be mathematically expressed as 
. M is the monthly cost, C is the number of cars owned, and k is the constant of variation.
Given that it costs $420 a month to insure 3 cars, we can find the k-value.
Divide both sides by 3.
Now, we have the mathematical relationship.
Finding how much it costs to insure 5 cars can be found by substituting 5 for C and solving for M.
The statement, 'The monthly costly to insure your cars varies directly with the number of cars you own' can be mathematically expressed as
Given that it costs $420 a month to insure 3 cars, we can find the k-value.
Divide both sides by 3.
Now, we have the mathematical relationship.
Finding how much it costs to insure 5 cars can be found by substituting 5 for C and solving for M.
← Didn't Know|Knew It →
If the roots of a function are 
, what does the function look like in 
format?
If the roots of a function are
Tap to reveal answer
This is a FOIL problem. First, we must set up the problem in a form we can use to create the function. To do this we take the opposite sign of each of the numbers and place them in this format: 
.
Now we can FOIL.
First: 
Outside: 
Inside: 
Last: 
Then add them together to get 
.
Combine like terms to find the answer, which is 
.
This is a FOIL problem. First, we must set up the problem in a form we can use to create the function. To do this we take the opposite sign of each of the numbers and place them in this format:
Now we can FOIL.
First:
Outside:
Inside:
Last:
Then add them together to get
Combine like terms to find the answer, which is
← Didn't Know|Knew It →
Tom is painting a fence 
feet long. He starts at the West end of the fence and paints at a rate of 
feet per hour. After 
hours, Huck joins Tom and begins painting from the East end of the fence at a rate of 
feet per hour. After 
hours of the two boys painting at the same time, Tom leaves Huck to finish the job by himself.
If Huck completes painting the entire fence after Tom leaves, how many more hours will Huck work than Tom?
Tom is painting a fence
If Huck completes painting the entire fence after Tom leaves, how many more hours will Huck work than Tom?
Tap to reveal answer
Tom paints for a total of 
hours (2 on his own, 2 with Huck's help). Since he paints at a rate of 
feet per hour, use the formula
(or 
)
to determine the total length of the fence Tom paints.
feet
Subtracting this from the total length of the fence 
feet gives the length of the fence Tom will NOT paint: 
feet. If Huck finishes the job, he will paint that 
feet of the fence. Using 
, we can determine how long this will take Huck to do:
hours.
If Huck works 
hours and Tom works 
hours, he works 
more hours than Tom.
Tom paints for a total of
to determine the total length of the fence Tom paints.
Subtracting this from the total length of the fence
If Huck works
← Didn't Know|Knew It →
Two numbers have a ratio of 5:6 and half of their sum is 22. What are the numbers?
Two numbers have a ratio of 5:6 and half of their sum is 22. What are the numbers?
Tap to reveal answer
Set up the equation:
Solve the equation:
Find the two numbers:
The two numbers have a ratio of 5:6, therefore the ratio can also be represented as:
The two numbers are 20 and 24.
Set up the equation:
Solve the equation:
Find the two numbers:
The two numbers have a ratio of 5:6, therefore the ratio can also be represented as:
The two numbers are 20 and 24.
← Didn't Know|Knew It →
What is the equation of the line that has a slope of 3 and passes through the point (3,-6)?
What is the equation of the line that has a slope of 3 and passes through the point (3,-6)?
Tap to reveal answer
The equation for a line in slope-intercept form is:
where 
and 
are the known coordinates (3,-6).
Substituting gives
,
and simplifying gives the final answer:
The equation for a line in slope-intercept form is:
where
Substituting gives
and simplifying gives the final answer:
← Didn't Know|Knew It →
Set up an equation that properly displays the information given.
Suzanne has a pack of multi-colored jelly beans. She wants to sort them into equal amounts to give out to her four friends, but not until she eats eight of them. If the total pack contains 60 jelly beans, then how many is each friend going to get?
Set up an equation that properly displays the information given.
Suzanne has a pack of multi-colored jelly beans. She wants to sort them into equal amounts to give out to her four friends, but not until she eats eight of them. If the total pack contains 60 jelly beans, then how many is each friend going to get?
Tap to reveal answer
Let 
be the number of jelly beans that each friend will receive. She has four friends, so the total number of jelly beans her friends will receive is 
. Suzanne eats another eight, so the equation can be written as 
.
Let
← Didn't Know|Knew It →
A circular tower stands surrounded by a circular moat. A bridge provides a passage over the moat to the tower. The distance from the outer edge of the moat to the center of the tower is 
meters. The area of the floor of tower is 
. How long is the bridge over the moat?
A circular tower stands surrounded by a circular moat. A bridge provides a passage over the moat to the tower. The distance from the outer edge of the moat to the center of the tower is
Tap to reveal answer
The distance from the outer edge of the moat to the center of the castle is the radius (100 m) of the larger circle formed by the outer edge of the circular moat.
The radius of the tower's floor (found using the area of the floor), needs to be subracted from 100 m to find the distance fo the bridge.
The distance from the outer edge of the moat to the center of the castle is the radius (100 m) of the larger circle formed by the outer edge of the circular moat.
The radius of the tower's floor (found using the area of the floor), needs to be subracted from 100 m to find the distance fo the bridge.
← Didn't Know|Knew It →
Your friend goes on a diet to lose a little weight. He starts at 
pounds and cuts his calories by 
a day. Write a linear equation to express your friend's weight in pounds as a function of weeks on the diet. Hint: there are 
,
calories in a pound.
Your friend goes on a diet to lose a little weight. He starts at
Tap to reveal answer
The question asks for a relation between pounds lost and weeks on the diet. If each day your friend cuts 500 calories, the number of pounds he is losing per week is 1:
The rate of change, or slope, is therefore -1. The slope is negative because the independent variable (weight in pounds) is decreases as the dependent variable (time in weeks) increases. The y-intercept is 180, because that is how much your friend weighs at the start, when time = 0. Plugging these values into 
form, we end up with:
The question asks for a relation between pounds lost and weeks on the diet. If each day your friend cuts 500 calories, the number of pounds he is losing per week is 1:
The rate of change, or slope, is therefore -1. The slope is negative because the independent variable (weight in pounds) is decreases as the dependent variable (time in weeks) increases. The y-intercept is 180, because that is how much your friend weighs at the start, when time = 0. Plugging these values into
← Didn't Know|Knew It →
Solve for 
in the following equation: 
Solve for
Tap to reveal answer
Starting with the equation 
, you want to collect like terms.
Put all of the numbers on one side, and leave only the variable on the other side.
The first step is to subtract 
from both sides.
You get 
.
The next step is to divide both sides by 
to get the final answer, 
.
Starting with the equation
Put all of the numbers on one side, and leave only the variable on the other side.
The first step is to subtract
You get
The next step is to divide both sides by
← Didn't Know|Knew It →
Express as an equation.
more than 
is 
Express as an equation.
Tap to reveal answer
Take every word and translate into math.
more than means that you need to add 
to something which is 
.
Anytime you see "is" means equal.
Now let's combine and create an expression of 
Take every word and translate into math.
Anytime you see "is" means equal.
Now let's combine and create an expression of
← Didn't Know|Knew It →
Express as an equation.
times 
is 
less than 
Express as an equation.
Tap to reveal answer
Take every word and translate into math.
times something means that you need to multiply 
to something which is 
.
less than means that you need to subtract 
from 
.
Anytime you see "is" means equal.
Let's combine to get 
.
Take every word and translate into math.
Anytime you see "is" means equal.
Let's combine to get
← Didn't Know|Knew It →
If Bob's age is 
years old and Jack is 
more than 
times Bob's age, then express Jack's age in terms of 
.
If Bob's age is
Tap to reveal answer
Take every word and translate into math.
more than means that you need to add 
to something.
times something means that you need to multiply 
to Bob's age which is 
.
Now we can combine them to have an expression of 
.
Take every word and translate into math.
Now we can combine them to have an expression of
← Didn't Know|Knew It →
There are 
marbles in a jar. There are 
types of color: red, blue, green and yellow. There are 
red marbles, 
blue marbles, 
green marbles. Find an equation to represent the number of yellow marbles.
There are
Tap to reveal answer
We have 
known values and 
unknown value. We have a total and that these 
colors add up to 
.
Let's represent the colors with variables corresponding to the first letter of the color. 
.
Now, plug in the values that are known. Final answer is 
.
To simplify we add the constants which results in:
We have
Let's represent the colors with variables corresponding to the first letter of the color.
Now, plug in the values that are known. Final answer is
To simplify we add the constants which results in:
← Didn't Know|Knew It →
Express as an equation.
Difference between 
times 
and the quotient of 
and 
is 
more than 
times 
.
Express as an equation.
Difference between
Tap to reveal answer
Take every word and translate into math. What a difference means, is that 
is the first number subtracting 
.
The 
part is 
times something means that you need to multiply 
to something which is .
The 
part is quotient.
Anytime you take a quotient of 
and 
, 
is the in the numerator and 
is in the denominator. Therefore the expression is 
.
Anytime you see "is" means equal.
more than means that you need to add 
to something.
That something is 
multipled by 
or 
.
Let's just combine them to have an expression of 
.
Take every word and translate into math. What a difference means, is that
The
The
Anytime you take a quotient of
Anytime you see "is" means equal.
That something is
Let's just combine them to have an expression of
← Didn't Know|Knew It →