Semester 1 Flashcards : Tanvi Vardhini Mahesh Vidya

Find the solution for the following set using substitution y = 4x y = 7x - 30

( 10, 40 )

Find the solution for the following set using the graphing method y = (1/4)x -4 y = (-1/2)x -1

( 4, -3 )

Find the solution for the following set using elimination. 7x + 2y = 27 9x + 2y = 37

( 5, -4 )

x + 5 = x - 3

5 = -3 , No Solution

am = n + p , solve for a

a = (n+p)/m , In this problem to get "a" by itself, you would divide both sides by m.

Stitches Inc. can make at most 30 jean jackets and 20 leather jackets in a week. It takes a worker 10 hours to make a jean jacket and 20 hours to make a leather jacket. The total number of hours by all of the employees can be no more than 500 hours per week. The profit on a jean jacket is $20, and the profit on a leather jacket is $50. (a) What are your variables and what do they represent? (b) Write the objective quantity equation. (c) Write the system of the inequalities that describes the constraints. (d) Graph the system inequalities and find the vertices. (e) How man of each type should be produced in order to maximize their profit? What is the maximum Profit?

(a) : x is the number of jean jackets, y is the number of leather jackets. (b) : f(x,y) = 20x + 50y (c) x ≥ 0, y ≥ 0, 10x + 20y ≤ 500, x ≤ 30, y ≤ 20 (d) (0,0) , (10,20) , (0,20) , (30,10) , (30,0). (e) 10 jean jackets and 20 leather jackets. The maximum profit is $1200.

Evaluate the following expression given the function below: f(x) = -3x + 1 , find f(10)

-29

Find the 55th term of the arithmetic sequence 11, 30, 49....

1037

The sum of three consecutive integers is 54. Find the integers.

17, 18 and 19

How much of a 60% alcohol solution must be mixed with 3 cups of a 20% alcohol solution to create a 40% alcohol solution.

3 cups

Literal equations

A formula or equation with several variables. They will never say a variable equals a number. It consists primarily of variables.These type of equation usually end up with variables on both sides. They cannot be simplified completely or they will have variables on both sides. It will always tell you to solve for a variable, so you would try and get that variable on one side, so that it says the variable that they asked for equals a specific equation. In literal equations they are 2 types of them, multi-step and single-step.

Functions and Function notation

A function is a relationship between an input (x-value) and an output ( y-value) in which there is only one output for each input, y-values can be repeated but x-values cannot be repeated. A relation is a set of ordered pairs, a domain is the set of all possible x coordinates (input), a range is the set of all possible y coordinates (output). Functions can be written as function notations. The f (x) notation is another way of representing the y-value in a function, y = f (x), here f means function, so x in f (x) is the x value, f (x) means function of x or in other words y equals.

Linear Programming

A method for finding a minimum or maximum value of some quantity, given a set of constraints. Linear programming is a technique for a system of linear constraints and a linear objective function. An objective function defines the quantity to be used, and the goal of linear programming is to find the values of the variables that maximize or minimize the objective function. The important thing is that the maximum and minimum had to satisfy the constraints or requirements.

Find the Correlation Coefficient using Desmos from the points, round to the nearest thousandth place. Determine if it is positive or negative and if it it weak or strong. (2,6) , (7,4) , (5,1) , (2,2) , (10,1) , (12,3)

Correlation Coefficient is -0.316. It is weak negative correlation.

Standard deviation

In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range. Standard Deviation is a measure of variability that describes an average distance of every score from the mean. In simpler words Standard Deviation is the mean of the mean. To find it by hand first find the mean, then for each number: subtract the mean, square the result, then add up all of the squared results, then divide this sum by one less than the number of data points (N - 1), Take the square root of this value to obtain the sample standard deviation. To find the standard deviation using desmos is to make a table and put the values in for the x1 column, then click on the keypad on the bottom left most corner, then click on functions and click on stats, then click on stdev, then in the parenthesis, put x1 to find the standard deviation or you do not have to make the table, you can just click on the keypad, then functions, then stats, then stdev and put the values/numbers in the parenthesis.

Systems of Equations graphing

One of the ways to solve systems of equations is to graph. In math, graphing means to graph both the equations and find a intersection point and that point is your solution. This point will satisfy both the equations, i other words it will make both the equations true. Step 1 is to make sure both the equations are in slope-intercept form, because it is easier to graph when you have an equation that is in slope-intercept form, Step 2 is to graph both the equations, Step 3 is to find the intersection, and that is your solution, Step 4 is to organize solution in (x,y) format.

Find the standard deviation of 41, 46, 49, 52, 57, to the thousandth place using desmos and explain it.

The standard deviation is 6.042. This means that this data has numbers that mostly lie between 6.042 values from the mean which is 49.

Equation with infinitely many solutions

There are a lot of solutions, they are infinite solutions. In this type of any number/ a set of numbers will always make the equation true. When they are simplified they look the same on both sides.

Solve and graph the inequalities -4(3 - x) > 5(x + 1)

x < -17

(2x + 4)/ 2 = x + 2

x = x , this means any number makes this equation true.

Write the slope-intercept form for an equation of the line that passes through (6,4) and is perpendicular to the line of y = 3x - 8

y = (-1/3)x + 6

Find the slope and y-intercept from the following points, (2,6) and (4,7) and put it in slope-intercept form.

y = (1/2)x + 5

Find the Regression line using Desmos from the points, round to the nearest thousandth place. (2,6) , (7,4) , (5,1) , (2,2) , (10,1) , (12,3)

y = -0.148x + 3.773

Find the Residuals using Desmos from the points, round to the nearest thousandth place if you have to. (2,6) , (7,4) , (5,1) , (2,2) , (10,1) , (12,3)

Residual for (2,6) : 2.523 Residual for (7,4) : 1.266 Residual for (5,1) : -2.031 Residual for (2,2) : -1.477 Residual for (10,1) : -1.289 Residual for (12,3) : 1.008

Residuals

Residuals are the distance from the point to the line. They are sometimes positive and sometimes negative. If you want to find out if it is accurate, then add up all the residuals and they should equal 0 or close to 0. The residual is the observed value - predicted value. If you want to find t by hand, first find the predicted value you would plug in the x value in the equation of the regression line and solve to get a number. Then you would subtract this from the y value to find the residual. If you want to do it in desmos, first click on the plus sign on the left, then click on the table and enter all the points in. Then go to the next box and enter this equation : y_1~ mx_1 + b, if you do this desmos will give you the Residuals, if you click e1 plot, it will add another column to the table and it will tell you the residual for each point.

3x + 10 < 15 : Put this Inequality on a Coordinate Grid

See the graph (Graph made in Desmos )

Solve and Graph : 2x-y > -3

See the graph (Graph made in Desmos )

The sum of three odd consecutive integers is 69. Find the integers.

21, 23 and 25

3x - 5 = 2x

5 = x, the only value that makes this equation true is 5.

Arithmetic sequences

Arithmetic sequences are mathematical sequences that have the same difference. It is a sequence in which each consecutive stage has a common difference, when they are like this, they are called linear, or arithmetic. They can be solved using a list, equation or a table or a graph. The explicit equation is an=a1+(n-1)d

Standard form

Ax + By = C, This equation is called standard form. Using what we know we can conclude and agree that this is another way to write the equation of a line. In standard form, "A" cannot be a negative number, "A,B and C" must be integers, they cannot be fraction or decimals. "B and C" can be negative. To find the slope from standard form you would put -A/B and to find the y-intercept you would put C/B. The use of the standard form is that when an equation is given in this form, it's pretty easy to find both intercepts (x and y). This form is also very useful when solving systems of two linear equations.

Composite functions

Combination of two functions, where you apply the first function and get an answer, and then fill that answer into the second function. They usually give you f(x), g(x), h(x), j(x), and so on. So to start with they will give you an equation for f (x), and they will give you a value for x and you would have to plug it in for the equation that has f (x) =, and solve it to get an final answer. If the question says f(g(x)) or some thing like that first you would solve the equation for g(x), then put that value in the equation for f(x) and solve to get your final answer.

Consecutive even/odd word problems

Consecutive integer even/odd problems are word problems that involve even or odd consecutive integers or numbers that skip numbers in between that are even or odd. They usually have a number that they equal, and the question will most likely ask you to find the consecutive integers.Since the sum of an even number and an odd number is always odd, we always add 2, 4 and so on. If we think about, you can realize that even numbers can get one even/odd number to another even/odd number. First define the variables, then set up an equation, then solve the equations to find the integers, then check your work.

Consecutive integer word problems

Consecutive integer problems are word problems that involve consecutive integers or numbers that go in order. They usually have a number that equals the consecutive numbers, and the question will most likely ask you to find the consecutive integers. To solve consecutive integer word problems, first define your variables, to begin with ,make x your first variable to begin with, for example, defining a term (X) as 1st number, another term as 2nd number, and another term as the 3rd and so on until you have defined all your variables.Then set up an equation to go with your word problem (Think : Translating Expression! ). Then solve your equation. Find the integers and remember , "x" is the first integer/term. Check your work in the end.

Correlation Coefficient

Correlation Coefficient is a number between -1 and 1. When you are close to 1 it is strong positive correlation , when it is closer to 0, it is weak positive correlation, or sometimes even no correlation when it is really close to 0. When you are close to -1 it is strong negative correlation , when it is closer to 0, it is weak negative correlation, or sometimes even no correlation when it is really close to 0. The best and easy way to find the Correlation Coefficient is to do it using desmos, if we have to do it by hand, we can only make a educated guess. If you want to do it in desmos, first click on the plus sign on the left, then click on the table and enter all the points in. Then go to the next box and enter this equation : y_1~ mx_1 + b, if you do this desmos will give you the Correlation Coefficient.

Equation with no Solution

Equation that have no solution not even zero because that is a solution. In this type of equations, you end up with no variables, this happens because you have canceled out all the variables and you final answer is a number equals a number, but this make the equation false. "No solution" means that there is no value, not even 0, which would satisfy the equation.

Function Problem : Find out whether these points are a function. {(6,1),(8,2),(−4,5),(6,3)} Function Notation Problem : Translate the following statement into coordinate points : f ( 2 ) = 7

Function Problem : Not a Function Function Notation Problem : (2,7)

Parallel lines

In geometry, parallel lines are lines in a plane which do not meet; that is, two straight lines in a plane that do not intersect at any point are said to be parallel and they always maintain a distance. Parallel lines on a graph have the same slope but different y-intercepts. This is the reason the never meet. If you want to find out if two equations make parallel lines or not, you look at the slopes, if they are same they are parallel, if they are not the same, they are not parallel.

Graphing Systems of Inequalities

In order to graph systems of inequalities, first write the inequality as an equation. Put it in slope and y-intercept format, graph the line, find out if the line is dashed or solid. Test a point that is not on the line to see if it makes a true statement. If (0,0) is not on the line, it is the easiest point to test. If true shade on the side of the line that contains the test point. If false, shade on the side of the line does not contain the test point. The solution is the shaded area or the feasible region on a graph.

Inequalities on a Coordinate Grid

Inequalities on a coordinate grid help see or visualize or understand the the inequality. When the inequality has x is <,>,≤,≥, a vertical line is being used. When the inequality has y is <,>,≤,≥ a horizontal line is being used. When a < or > is used, we have to draw a dotted line and when ≤ or≥ is used, we have to draw a solid line.(The following is true for simple inequalities like x > 5 or y ≥ 7) When x is on left and next to it is < or ≤ , it is shaded towards the left. When x is on left and next to it is > or ≥, it is shaded towards the right. When y is on left and next to it is < or ≤ , it is shaded towards the top. When y is on left and next to it is > or ≥, it is shaded towards the bottom. When there is a inequality that says 2x + 3 < 9, then the line is slanted.

Simple Inequalities

Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own, but when you multiply or divide by a negative number you change the sign by simply flipping it over or changing the direction it faces. To solve you add or subtract first to try and get the variable by itself, then you would multiply or divide to get the variable by itself and To graph the equation on a number line, first you would find if you need a closed circle or an open circle. When the symbol is < or >, it is a open circle. When the symbol is ≤ or ≥ , the circle is filled in /the closed circle. In the end to draw the line, if the symbol < or ≤ has the variable on the right, the line will go to the right but if the variable was on the left you would draw the line towards the left. When the symbol is > or ≥ and the variable is on the right, you would draw the line towards the left side and if the variable was on the left you would draw the line towards the right.

Mixture Word Problems

Mixture problems involve creating a mixture from two or more things, and then determining some quantity (percentage, price, etc) of the resulting mixture. These type of questions usually have 2 types of material/item and they are being mixed and you will be asked to find a missing quantity or percentage. The best way to solve these type of questions is to set up a table, then solve.Then set up equations to go with your table and solve.

Systems of equations elimination

One of the ways to solve systems of equations is to eliminate. In math, elimination means to get rid of a variable because it has the same absolute value as its coefficient, for example you can eliminate 2x and 2x (here you would have to multiply one of the equations by -1 or any other negative number to make sure they can be eliminated) or 5y and -5y. Step 1 is to Subtract to eliminate a variable, Step 2 is to Solve for the remaining variable, Step 3 is to Substitute the known value into an original equation to solve for the second variable, Step 4 is to organize solution in (x,y) format,

Systems of equations substitution

One of the ways to solve systems of equations is to substitute. In math, substitution means replacing a variable or a letter with a number or an expression, step 1 is to Set the equations equal to one another, Step 2 is to Solve for the variable you are solving for which most likely is going to be x. Step 3 is to Substitute the known value into an original equation to solve for the second variable, which is most likely y. Step 4 is to organize solution in (x,y) format.

Find the slope and y-intercept from the following equation. 4x - 2y = -6

Slope : 2 , Y-intercept : 3

Determine if the equations are parallel, perpendicular or neither. 6y = -18x + 8 and 12x + 4y =10

Parallel

Perpendicular lines

Perpendicular lines are two or more lines that intersect at a 90-degree angle.Two lines are perpendicular if and only if their slopes are negative reciprocals. To find out if two equations are perpendicular or not, first to see if the slopes are the opposite reciprocal of the slope of the first equation, the y-intercepts should be different. If all this applies, both the equations are perpendicular to each other, if it does not apply, they are not perpendicular.

Regression line

The Regression line is also called the line of best fit. A regression line is a line best used to represent the data in a scattered plot, also known as the trend line or linear regression line. In order to make your regression line accurate, make sure it splits your scatter plot accurately in half, or in other words, same number of dots on each side. If you want to find it by hand, draw a line diving the dots in half and make sure there are the same amount of dots on both side. Then find the slope and y-intercept to find the equation, or you can work backwards. If you want to find it in Desmos, first click on the plus sign on the left, then click on the table and enter all the points in. Then go to the next box and enter this equation : y_1~ mx_1 + b, if you do this desmos will give you the slope and y-intercept, and then you can write the line of best fit.

Slope-intercept form

The slope-intercept form is simply the way of writing the equation of a line so that the slope (steepness) and y-intercept (where the line crosses the vertical y-axis) can be seen and understood easily. The right way to right in slope-intercept form is to use y = mx + b, where m is the slope, which can be found by subtracting the 1st y value from the 2nd y value and then subtracting the 1st x value from the 2nd x value and diving difference in y over the difference in x. The b represents the y intercept, this can be found in the graph or by substituting the x and y value to find the y-intercept. You can find the slope like this when it is on a table or when the points are given because they both show you the x and y values. The x and y values can be found from the points. To find the slope and y-intercept on a graph from a graph, you would do rise over run, to do this you pick two points on a graph and see how many values it goes up/down, then you would find the run by seeing how many values to the right/left it is. Make sure to make sure if the slope is positive or negative. Then to find the y-intercept you would see where the line crosses on the y-intercept and that is the y-intercept.

Equation with one solution

They only have one solutions. In these type of equations, you end up with x or the variable equals some number. Only one number/value makes the equation true, and all the other numbers/values would make the equation false.