Patterns and generalisations

Coordinate geometry basics

Linear equations

Systems of linear equations

Linear modelling

MYP Standard Mathematics Systems of Linear Equations Notes

Solutions Mean "Where The Conditions Match"

Each linear equation describes a straight line when graphed.

Solving the system is equivalent to finding where the lines intersect.

If a point lies on a line, it satisfies that equation.
If a point lies on both lines, it satisfies the system.

Analogy

Think of each equation as a "rule".
A solution to the system is like an answer that follows both rules at once.

A System Of Two Lines Has Only Three Possible Outcomes

When you graph two linear equations, there are only three solution scenarios.

Definition

Unique solution

Exactly one ordered pair $(x,y)$ satisfies both equations. Graphically, the two lines intersect at one point.

Definition

No solution

No ordered pair satisfies both equations. Graphically, the lines are parallel (same gradient, different intercepts).

Definition

Infinitely many solutions

Infinitely many ordered pairs satisfy both equations. Graphically, the lines are coincident (the same line written in two forms).

How To Recognize The Three Cases From Equations

For lines in the form $y=mx+b$:

Unique solution when gradients are different ($m_1 \ne m_2$).

No solution when gradients are equal but intercepts differ ($m_1=m_2$ and $b_1\ne b_2$).

Infinitely many solutions when both gradient and intercept match ($m_1=m_2$ and $b_1=b_2$), meaning the equations are equivalent.

Note

A system of two linear equations cannot have "two different unique solutions."
Two straight lines cannot intersect in two separate points.

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Note

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Tip

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A system of linear equations is a set of two (or more) linear equations that must be true at the same time.
The goal is to find the ordered pair (or pairs) $(x,y)$ that satisfy every equation in the system.

Definition

System of linear equations

A set of two or more linear equations involving the same variables, solved together to find values that satisfy every equation in the system.

Solutions Mean "Where The Conditions Match"

Each linear equation describes a straight line when graphed.
Solving the system is equivalent to finding where the lines intersect.
1. If a point lies on a line, it satisfies that equation.
2. If a point lies on both lines, it satisfies the system.

Analogy

Think of each equation as a "rule".
A solution to the system is like an answer that follows both rules at once.

A System Of Two Lines Has Only Three Possible Outcomes

When you graph two linear equations, there are only three solution scenarios.

Definition

Unique solution

Exactly one ordered pair $(x,y)$ satisfies both equations. Graphically, the two lines intersect at one point.

Definition

No solution

No ordered pair satisfies both equations. Graphically, the lines are parallel (same gradient, different intercepts).

Definition

Infinitely many solutions

Infinitely many ordered pairs satisfy both equations. Graphically, the lines are coincident (the same line written in two forms).

How To Recognize The Three Cases From Equations

For lines in the form $y=mx+b$:

Unique solution when gradients are different ($m_1 \ne m_2$).
No solution when gradients are equal but intercepts differ ($m_1=m_2$ and $b_1\ne b_2$).
Infinitely many solutions when both gradient and intercept match ($m_1=m_2$ and $b_1=b_2$), meaning the equations are equivalent.

Note

A system of two linear equations cannot have "two different unique solutions."
Two straight lines cannot intersect in two separate points.

Graphical Solution: Intersection As The Answer

A graphical method is a powerful way to understand what is happening: you graph both equations and read off the intersection point.

MYP Standard Mathematics Systems of Linear Equations Notes

Solutions Mean "Where The Conditions Match"

A System Of Two Lines Has Only Three Possible Outcomes

How To Recognize The Three Cases From Equations

Graphical Solution: Intersection As The Answer

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Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

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Solutions Mean "Where The Conditions Match"

A System Of Two Lines Has Only Three Possible Outcomes

How To Recognize The Three Cases From Equations

Graphical Solution: Intersection As The Answer

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

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Number and Algebra Foundations7 subtopics

Coordinate Geometry and Linear Modelling5 subtopics

Functions and Quadratic Relationships4 subtopics

Sets, Logic and Probability4 subtopics

Geometry of Shapes, Surface Area and Volume3 subtopics

Statistical Measures and Data Representation3 subtopics

Right Triangles and Circle Geometry3 subtopics

Sequences, Formulae and Proportional Reasoning3 subtopics

Sampling Methods and Bivariate Data2 subtopics

MYP Standard Mathematics Systems of Linear Equations Notes

Number and Algebra Foundations7 subtopics

Coordinate Geometry and Linear Modelling5 subtopics

Functions and Quadratic Relationships4 subtopics

Sets, Logic and Probability4 subtopics

Geometry of Shapes, Surface Area and Volume3 subtopics

Statistical Measures and Data Representation3 subtopics

Right Triangles and Circle Geometry3 subtopics

Sequences, Formulae and Proportional Reasoning3 subtopics

Sampling Methods and Bivariate Data2 subtopics

Solutions Mean "Where The Conditions Match"

A System Of Two Lines Has Only Three Possible Outcomes

How To Recognize The Three Cases From Equations

Graphical Solution: Intersection As The Answer

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Solutions Mean "Where The Conditions Match"

A System Of Two Lines Has Only Three Possible Outcomes

How To Recognize The Three Cases From Equations

Graphical Solution: Intersection As The Answer

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account