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3-Variable Systems: No Solution vs Infinite Solutions

Grade 12 Math: Master Advanced Concepts & Skills

Analyze puzzles and games that involve numerical and logical reasoning, using problem-solving strategies

Factor by taking out the greatest common factor

Factor by taking out GCF

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Factor by grouping

Factoring difference of squares: \(x^2 - y^2\)

Factoring difference of squares: <katex>x^2 - y^2</katex>

Factoring difference of squares: x^2 - y^2

Factoring trinomials

Factoring Trinomials

Difference quotient: applications of functions

Solve problems that involve quadratic equations

Solving absolute value equations

Solving absolute value inequalities

Solving Absolute Value Inequalities Involving "less than"

Solving Absolute Value Inequalities Involving "greater than"

Recognizing Absolute Value Inequalities with "No Solution" or "All Real Numbers"

Graphing reciprocals of linear functions

Graphing reciprocals of quadratic functions

Determine equivalent forms of rational expressions (limited to numerators and denominators that are monomials and binomials)

Solving 3 variable systems of equations by substitution

Solving rational inequalities

Solving Rational Inequalities With Two Fractions

System of linear equations

System of linear-quadratic equations

System of quadratic-quadratic equations

Perform operations on rational expressions (limited to numerators and denominators that are monomials and binomials)

Solving 3 variable systems of equations by elimination

Word problems relating 3 variable systems of equations

Solving 3 variable systems of equations (no solution, infinite solutions)

System of Equations With No Solution

System of Equations With Infinite Solutions

System of Equations With Infinite Solutions - Extended

Graphing linear inequalities in two variables

What is a logarithm?

Logarithms and Exponential Growth

Understanding Logarithms through Exponential Growth

Understanding Logarithms

Converting from logarithmic form to exponential form

Evaluating logarithms without a calculator

Common logarithms

Natural log: ln

Product rule of logarithms

Quotient rule of logarithms

Power rule of logarithms

Solve problems that involve rational equations (limited to numerators and denominators that are monomials and binomials)

Graphing quadratic inequalities in two variables

Graphing systems of quadratic inequalities

Solving exponential equations with logarithms

Exponential growth and decay by a factor

Exponential decay: Half-life

Exponential growth and decay by percentage

Finance: Compound interest

Finance: Compound Interest

Continuous growth and decay

Graphing systems of linear inequalities

Applications of inequalities

Solving exponential equations using exponent rules

Demonstrate an understanding of logarithms and the laws of logarithms

What is linear programming?

Finding Optimal Values Given Constraints In Slope-Intercept Form

Finding Optimal Values Given Constraints In Standard Form

Linear programming word problems

Graphing exponential functions

Graphing transformations of exponential functions

Graphing logarithmic functions

Graphing transformations of logarithmic functions

Logarithmic scale: Richter scale (earthquake)

Logarithmic scale: pH scale

Logarithmic scale: dB scale

Angle in standard position

Coterminal angles

Coterminal Angles

Solve problems that involve exponential equations

Unit circle

Polynomial functions

Recognizing a Polynomial Function

Characteristics of polynomial graphs

Determining the equation of a polynomial function

Applications of polynomial functions

Multiplicities of polynomials

Reference angle

Find the exact value of trigonometric ratios

ASTC rule in trigonometry (All Students Take Calculus)

ASTC rule in trigonometry

Solving polynomial inequalities

Represent data, using exponential and logarithmic functions, to solve problems

Introduction to bearings

Law of sines

Ambiguous case: SSA triangles

Law of cosines

Applications of the sine law and cosine law

Sine graph: y = sin x

Cosine graph: y = cos x

Graphing transformations of trigonometric functions

Determining trigonometric functions given their graphs

Represent data, using polynomial functions (of degree ≤ 3), to solve problems

Bearings and direction word problems

Bearings and Direction Word Problems

Angle of elevation and depression

Arithmetic sequences

Arithmetic series

Arithmetic series formula

Represent data, using sinusoidal functions, to solve problems

Geometric sequences

Geometric series

Geometric Series

Infinite geometric series

Sigma notation

Research and give a presentation on a current event or an area of interest that involves mathematics

Finding limits from graphs

Continuity

Discontinuities of Rational Functions (denominator=0)

Classification of data

Characteristics of Quantitative Data

Classify the Level of Measurement

Identify the Level of Measurement

Arithmetic mean vs. Geometric mean

Solve problems that involve the application of set theory

Factoring difference of cubes

Factoring sum of cubes

Set notation

Understanding How to Use Set Notation

Set builder notation

Intersection and union of 2 sets

Intersection and Union of 2 Sets

Intersection and union of 3 sets

Intersection and Union of 3 Sets

Introduction to quadratic functions

Transformations of quadratic functions

Interpret and assess the validity of odds and probability statements

Characteristics of quadratic functions

Converting from general to vertex form by completing the square

Shortcut: Vertex formula

Converting general form into vertex form by applying the vertex formula

Vertex Formula Application

Ferris wheel trig problems

Tides and water depth trig problems

Spring (simple harmonic motion) trig problems

Completing the square

Completing the Square

Solve problems that involve the probability of mutually exclusive and non-mutually exclusive events

Finding the quadratic functions for given parabolas

Applications of quadratic functions

Solving quadratic equations by factoring

Solving quadratic equations by completing the square

Solve problems that involve the probability of two events

Using quadratic formula to solve quadratic equations

Solving a quadratic equation with TWO REAL SOLUTIONS

Solving a quadratic equation with TWO COMPLEX SOLUTIONS

Nature of roots of quadratic equations: The discriminant

Applications of quadratic equations

Solving quadratic inequalities

Solve problems that involve the fundamental counting principle

Solving polynomial equations by iteration

Fundamental counting principle

Fundamental Counting Principle Involving Restrictions

Path counting problems

Conversion between entire radicals and mixed radicals

Adding and subtracting radicals

Combining Like Radicals

Adding and Subtracting Radicals

Operations with radicals

We&apos;ve got you covered with our complete help for Grade 12 math, whether it&apos;s Ontario curriculum, <a href="https://www.alberta.ca/writing-diploma-exams.aspx">Diploma exam</a> (Alberta), <a href="http://www.edu.gov.mb.ca/k12/assess/tests/index.html">Provincial test</a> (Manitoba) or <a href="https://www.saskatchewan.ca/residents/education-and-learning/departmental-exams">Departmental exam</a> (Saskatchewan).

Aligned with your class or textbook, you will get grade 12 math help on topics like Trigonometric identities, Vectors, Permutations and Combinations, Matrices, Polynomial functions and so many more. Learn the concepts with our video tutorials that show you step-by-step solutions to even the hardest grade 12 math questions. Then, strengthen your understanding with tons of grade 12 math practice. 

All our lessons are taught by experienced Grade 12 math teachers. Let&apos;s finish your homework in no time, and ACE that final.

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Overview

No Solution or Infinite Solutions in 3-Variable Systems

What You'll Learn

Identify a no-solution system when elimination produces a false statement such as 0 equals 5

Identify an infinite-solutions system when elimination produces a true statement such as 0 equals 0

Apply the elimination method to reduce a 3-variable system down to a single equation or contradiction

Recognize dependent equations, where one equation is a multiple or combination of the others

Express an infinite solution set using a free parameter for one of the variables

Connect the algebra to geometry by picturing parallel planes or planes meeting along a shared line

What You'll Practice

Solving systems that result in false statements like 0 = 20

Identifying equations that are multiples of each other

Using multiplication and elimination on equation pairs

Determining whether a system has no solution or infinite solutions

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

3 Video lessons

Practice exercises

Learning resources

Skills

Systems of Equations

Three Variables

Elimination Method

No Solution

Infinite Solutions

Dependent Equations

Inconsistent Systems