Systems of linear equations Notes

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

Solutions Mean "Where The Conditions Match"

1. Each linear equation describes a straight line when graphed.
2. 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.

A System Of Two Lines Has Only Three Possible Outcomes

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

How To Recognize The Three Cases From Equations

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

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

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.

In this example, the lines $y=x+1$ and $y=4-2x$ intersect at $(1,2)$, so $(1,2)$ is the solution.

Algebraic Solution Methods: Substitution And Elimination

Graphing builds understanding, but algebraic methods give exact answers efficiently.

Substitution Works Best When A Variable Is Already Isolated

If one equation is already (or easily) written as $y=\dots$ or $x=\dots$, substitution is usually fastest.

Elimination Works Best When Coefficients Line Up

Elimination (also called addition method) aims to add or subtract equations to remove one variable.

Equivalent Transformations And Why Checking Matters

1. When solving, you often apply equivalence transformations (operations that keep the equation's solution set the same), such as adding the same value to both sides, multiplying both sides by a non-zero number, or expanding brackets.
2. Even if your algebra steps are valid, arithmetic mistakes happen.
3. That is why you should verify the final ordered pair in the original equations.

Modelling Real Situations With Systems Of Equations

Many real problems involve two conditions that must both be satisfied, so they naturally create a system.

A common modelling process is:

1. Identify variables and constraints (what is unknown, what must be true).
2. Translate each condition into an equation.
3. Solve the system.
4. Check in the originals.
5. Interpret the solution in context.

Mixture Problems (Nutrition Or Chemistry)

Mixtures are classic systems: one equation often represents a total amount (volume/mass), and another represents a total of some ingredient (acid, vitamins, etc.).

Systems Of Linear Inequalities: From A Point To A Region

1. A system of linear inequalities is similar, but instead of a single intersection point, the solution is usually a whole region of the plane.
   1. The boundary is a straight line.
   2. The solution set is one side of that line.
   3. The answer to a system is the overlap of the shaded regions.
2. Key graphing conventions:
   1. Use a solid boundary line for $\le$ or $\ge$ (boundary included).
   2. Use a dashed boundary line for $<$ or $>$ (boundary not included).
   3. Test a point (often $(0,0)$ if it is not on the boundary) to decide which side to shade.