2.1 Analytical Geometry- Mid point

2.1 Mid-point and Length of a Line Segment

1. What are coordinates of a point in the Cartesian plane?

The Cartesian plane, also known as the coordinate plane, is a two-dimensional plane formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). These lines are perpendicular and intersect at the origin, forming four quadrants.

A point in the Cartesian plane is represented by an ordered pair of numbers \((x,y)\), where ‘x’ is the distance from the point to the y-axis (x-coordinate), and ‘y’ is the distance from the point to the x-axis (y-coordinate).

2. What is the midpoint of a line segment?

The midpoint of a line segment is the point that divides the line segment into two equal segments. It is exactly halfway between the endpoints of the line segment.

1.6 System of Linear Equations

1.6 Solving Systems of Equations by Elimination Method

The method of elimination is a technique used to solve systems of linear equations. This method involves adding or subtracting the equations to eliminate one of the variables. Here’s a step-by-step process:

Step 1: Write Down the System of Equations

Write down the system of equations that you want to solve. For example, let’s consider the following system of equations:

\(2x + 3y = 12\)

\(4x - 3y = 6\)

1.5 System of equivalent Equations

1.5 Equivalent Equations

1.5.1) Equivalent Equations with Two Variables

Equivalent equations with two variables are equations that have the same set of solutions. For example, the equations \(2x + 3y = 6\) and \(4x + 6y = 12\) are equivalent because they represent the same line in the coordinate plane, and thus have the same set of solutions.

1.4 Solving system of linear equations by substitution

1.4 Solving by Substitution

To solve a system of linear equations by using substitution includes the following steps.

Consider we have system of equations as:

\(x + y = 10\) and \(2x + 3y = 8\)

Step 1: Isolate one of the variables in one of the equations

Choose one of the equations (any of the variables having coefficient as ‘1’) and solve it for one variable in terms of the other. This can be either x or y. For example, if you have the equation \(x + y = 10\), you can isolate \(x\) by subtracting \(y\) from both sides to get \(x = 10 - y\).

Quantum Numbers and Electronic Configuration of atoms

Writing Electronic Configuration of an Atom

1. Identify the atomic number (\(Z\)) of the atom. The atomic number is equal to the number of protons, which is also equal to the number of electrons in a neutral atom.
2. Fill the energy levels (shells) in order of increasing energy. Each energy level is designated by a principal quantum number (\(n\)) starting from 1, 2, 3, and so on.

3. Within each energy level, fill the subshells (\(s,p,d,f\)) in order of increasing energy. Each subshell is designated by an azimuthal quantum number (\(l\)) where \(s=0,p=1,d=2,and f=3\).
4. Each subshell can hold a certain number of electrons: \(s\) can hold 2, \(p\) can hold 6, \(d\) can hold 10, and \(f\) can hold 14.

1.3 Solving System of Equations by Graphing- U1L3MPM2D

1.3 Solving System of Equations by Graphing

Example: Solve the System of Equations by Graphing

Consider the following system of equations:

y = 2x + 4
y = -x + 1

Step 1: Identify the \(y-Intercept\) and Slope

Compare the given equations with the general slope-intercept form of a linear equation, \(y=mx+b\); '\(m\)' is the slope and '\(b\)' is the \(y-intercept\).

For the first equation, the \(y-intercept\) '\(b\)' is 4 and the slope '\(m\)' is 2. This means when \(x = 0\), \(y = 4\), and for every 1 unit increase in \(x\), \(y\) increases by 2 units.

For the second equation, the \(y-intercept\) ‘\(b\)’ is 1 and the slope '\(m\)' is -1. This means when \(x = 0\), y\( = 1\), and for every 1 unit increase in \(x, y\) decreases by 1 unit.

1.2 Graphing Linear System of Equations- U1L2MPM2D

Graphing Linear System of Equations

1) Graphing a Line

A line can be graphed by using following three methods:

• Using table of values
• Using slope and \(y-intercept\)
• Using \(x\) and \(y-intercepts\)