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 \(y-intercepts\)

1.1 Linear Equations- U1L1MPM2D

1.1 Solving Linear Equations

Step 1: Simplify Both Sides of the Equation

Before you start solving for the variable, make sure the equation is simplified as much as possible. This means:

• Combine like terms on each side of the equation.
• Apply the distributive property if necessary (i.e., expand brackets).

Step 2: Move Variable Terms to One Side

The goal is to isolate the variable on one side of the equation. If there are terms with the variable on both sides of the equation, use addition or subtraction to move all variable terms to one side.

Step 3: Move Constant Terms to the Opposite Side

Use addition or subtraction to move all constant terms (numbers without variables) to the opposite side of the equation from the variable.

Reciprocal Functions

Reciprocal of Linear and Quadratic Functions

The reciprocal of a function \(f(x)\) is given by \(g(x) = \frac{1}{f(x)}\).

Reciprocal of a Linear Function

Consider a linear function \(f(x) = mx + b\). The reciprocal is \(g(x) = \frac{1}{(mx + b)}\).

Graphing:

• If \(f(x) = 0\), then g(x) is undefined, resulting in a vertical asymptote.
• If \(f(x)\) is positive, then g(x) is also positive.
• If \(f(x)\) is negative, then g(x) is also negative.

Example

Let’s graph the reciprocal of the function \(f(x) = 2x + 1\).

• Identify the zeros of \(f(x): x = \frac{-1}{2}\) (vertical asymptote).
• Determine the sign of \(f(x)\): negative for \(x < \frac{-1}{2}\), positive for \(x > \frac{-1}{2}\).
• Sketch the graph of g(x) using this information.

Sampling and types of Sampling

1) Introduction to Sampling

1.1)            What is sampling?

Welcome to the "Sampling and Types of Sampling" course! In this chapter, we will be exploring the concept of sampling and its importance in various fields of study. Sampling is a fundamental technique used to gather data from a larger population in a systematic and efficient manner.

But what exactly is sampling? Let's break it down: