Comprehensive Algebra I Tutorial for Grade 9
This tutorial is designed for Grade 9 students to master key concepts in Algebra I. The topics covered include basic algebraic operations, solving equations, working with inequalities, graphing linear equations, and understanding quadratic equations. Each section contains examples, exercises, and solutions to ensure thorough understanding.
Outline
- Introduction to Algebra I
- Basic Algebraic Operations
- Solving Linear Equations
- Solving Inequalities
- Graphing Linear Equations
- Understanding Quadratic Equations
- Exercises with Solutions
- Homework
- Revision
Introduction to Algebra I
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In Algebra I, you will learn to work with variables, solve equations, and understand the properties of functions. This foundational knowledge is crucial for higher-level math and many real-world applications.
Basic Algebraic Operations
Algebraic operations include addition, subtraction, multiplication, and division of algebraic expressions. Understanding how to manipulate expressions is essential for solving equations.
Examples
Example 1:
Simplify: \(3x + 5x - 2x\).
Solution:
Combine like terms: \(3x + 5x - 2x = (3 + 5 - 2)x = 6x\).
Example 2:
Simplify: \(4(a - 2) + 3(2a + 5)\).
Solution:
Distribute and combine like terms:
\[ 4(a - 2) + 3(2a + 5) = 4a - 8 + 6a + 15 = 10a + 7. \]
Solving Linear Equations
To solve linear equations, isolate the variable on one side of the equation by using inverse operations.
Example 3:
Solve: \(2x - 3 = 7\).
Solution:
Add 3 to both sides: \(2x = 10\).
Divide by 2: \(x = 5\).
Example 4:
Solve: \(5(x - 1) = 3x + 7\).
Solution:
Expand and simplify: \(5x - 5 = 3x + 7\).
Move terms with \(x\) to one side and constants to the other: \(5x - 3x = 7 + 5\).
\(2x = 12 \implies x = 6\).
Solving Inequalities
Inequalities can be solved similarly to equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
Example 5:
Solve: \(3x - 7 \leq 5\).
Solution:
Add 7 to both sides: \(3x \leq 12\).
Divide by 3: \(x \leq 4\).
Graphing Linear Equations
To graph a linear equation, find the x-intercept and y-intercept, plot them on the coordinate plane, and draw a line through the points.
Example 6:
Graph: \(y = 2x + 3\).
Solution:
The y-intercept is 3 (point (0, 3)), and the slope is 2. Plot the y-intercept and use the slope to find another point (for example, (1, 5)). Draw a line through these points.
Understanding Quadratic Equations
A quadratic equation is of the form \(ax^2 + bx + c = 0\). It can be solved by factoring, completing the square, or using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Example 7:
Solve: \(x^2 - 5x + 6 = 0\).
Solution:
Factor the quadratic: \((x - 2)(x - 3) = 0\).
Set each factor to zero: \(x - 2 = 0\) or \(x - 3 = 0\).
Thus, \(x = 2\) or \(x = 3\).
Exercises with Solutions
Try solving these problems to test your understanding:
Exercise 1:
Simplify: \(7x + 2 - 4x + 6\).
Solution: Combine like terms: \(7x - 4x + 2 + 6 = 3x + 8\).
Exercise 2:
Solve: \(3(x + 2) = 9\).
Solution: Divide by 3: \(x + 2 = 3 \implies x = 1\).
Homework
Complete the following problems:
- Solve: \(4x + 7 = 19\)
- Simplify: \(2(x - 5) + 3(2x + 1)\)
- Graph the equation: \(y = -3x + 2\)
Revision
Review the examples and exercises to strengthen your understanding of these Algebra I concepts. Practice regularly to improve your skills.
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