This document discusses linear equations in two variables. It defines a linear equation as one that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0. It provides examples of linear equations and discusses how to graph them by plotting the x and y intercepts. It also explains how to determine if a given ordered pair is a solution to a linear equation by substituting the x and y values into the equation. Finally, it discusses different methods for solving systems of linear equations, including substitution and elimination.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
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I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
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Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
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The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
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The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
2. Linear Equations
Definition of a Linear Equation
A linear equation in two variable x is an
equation that can be written in the form ax +
by + c = 0, where a ,b and c are real numbers
and a and b is not equal to 0.
An example of a linear equation in x is .
3. Equations of the form ax + by = c are called
linear equations in two variables.
The point (0,4) is the y-intercept.
The point (6,0) is the x-intercept.
x
2
-2
This is the graph of the
equation 2x + 3y = 12.
(0,4)
(6,0)
4. Example:
Given the equation 2x + 3y = 18, determine if the
ordered pair (3, 4) is a solution to the equation.
We substitute 3 in for x and 4 in for y.
2(3) + 3 (4) ? 18
6 + 12 ? 18
18 = 18 True.
Therefore, the ordered pair (3, 4) is a solution to the
equation 2x + 3y = 18.
Solution of an Equation in Two Variables
5. The Rectangular Coordinate System
Each point in the rectangular coordinate system corresponds to an ordered pair of
real numbers (x,y). Note the word “ordered” because order matters. The first
number in each pair, called the x-coordinate, denotes the distance and direction
from the origin along the x-axis. The second number, the y-coordinate, denotes
vertical distance and direction along a line parallel to the y-axis or along the y-axis
Itself.
In plotting points,
we move across first
(either left or right), and
then move either up or
down, always starting
from the origin.
Point Movement from origin
(3.5) 3 units right and 5 units up
(4,-3) 4 units right and 3 units down
(-2,-7) 2 units left and 7 units down
(0,0) 0 right or left and 0 up or down
6. The Rectangular Coordinate System
In the rectangular coordinate system,
the horizontal number line is the x-axis.
The vertical number line is the y-axis.
The point of intersection of these axes is
their zero points, called the origin. The
axes divide the plane into 4 quarters,
called quadrants.
9. The Graph of an Equation
The graph of an equation in two variables is the set of
points whose coordinates satisfy the equation.
An ordered pair of real numbers (x,y)
is said to satisfy the equation when substitution of the x and y
coordinates into the equation makes it a true statement.
10. Let ax + by +c = O , where a ,b , c are
real numbers such that a and b ≠ O.
Then, any pair of values of x and y
which satisfies the equation ax + by +c
= O, is called a solution of it.
11. Find five solutions to the equation y = 3x + 1.
Start by choosing some x values and then computing the
corresponding y values.
If x = -2, y = 3(-2) + 1 = -5. Ordered pair (-2, -5)
If x = -1, y = 3(-1) + 1 = -2. Ordered pair ( -1, -2)
If x =0, y = 3(0) + 1 = 1. Ordered pair (0, 1)
If x =1, y = 3(1) + 1 =4. Ordered pair (1, 4)
If x =2, y = 3(2) + 1 =7. Ordered pair (2, 7)
Finding Solutions of an Equation
12. Plot the five ordered pairs to obtain the graph of y = 3x + 1
(2,7)
(1,4)
(0,1)
(-1,-2)
(-2,-5)
Graph of the Equation
14. • The method of substitution is not preferable if none of the
coefficients of x and y are 1 or -1. For example, substitution is
not the preferred method for the system below:
2x – 7y = 3
-5x + 3y = 7
• A better method is elimination by addition. The following
operations can be used to produce equivalent systems:
– 1. Two equations can be interchanged.
– 2. An equation can be multiplied by a non-zero constant.
– 3. An equation can be multiplied by a non-zero constant
and then added to another equation.
15. SUBSTITUTION METHOD:
The first step to solve a pair of linear equations by the
substitution method is to solve one equation for either of the
variables. The choice of equation or variable in a given pair does
not affect the solution for the pair of equations.
In the next step, we’ll substitute the resultant value of one
variable obtained in the other equation and solve for the other
variable.
In the last step, we can substitute the value obtained of the
variable in any one equation to find the value of the second
variable