tags:
  - algebra-1
  - secondary-level

Linear Equations

Learning checklist

Errors I often make

When multiplying out brackets, ensure the sign of the product/s is correct.
When moving terms across the equals sign, ensure the sign correctly reflects how you removed the term from the original side. E.g "+2" moved across the equals becomes "-2" because it was subtracted from the original side. Think: I'm not moving +2 across; I'm subtracting it from each side.

We call an equation linear if every term is either a constant or a product of a constant and the first power of a single variable.
Why we don't use x as the sign for multiplication; what we do instead; dots, parentheses, proximity.
Define term (a constant, or a variable and coefficient) and expression (terms separated by + or -)
Explain the various coefficients in for example 4xy (4 is c of xy; 4x is c of y; x is c of 4y )
Define like terms (same variable raised to the same power)
Combine like terms by keeping variables the same and performing operations on coefficients
Simplify an expression by removing parentheses and combining like terms
Define equation (a statement two algebraic expressions are equal; always includes the = symbol)
Test a solution for a variable in an equation by simple substitution (plug it in, see if it gives true result)
Understand the addition property of equality (we can add or subtract a value from both sides of an equation without changing the solution)
- Additive inverse property any number plus its negative is zero
- Additive identity of zero any number plus zero is unchanged.
In a simple equation solvable by addition and subtraction, isolate a variable by inverse operations (add or subtract from both sides until you get x = number)
Understand the multiplicative property of equality (we can mult or div both sides by a non-zero value without changing the solution)
Multiplicative identity property of 1 any number x1 is unchanged
Multiplicative invers property any non-zero number, expressed as fraction, x its reciprocal, is 1
In a simple equation solvable by mult and div, isolate a variable by inverse operations
Solve linear equations in one variable with a 4 step process.
Spot conditional (normal solvable, inc. 1y = 0), identity (0y = 0 but not 1y = 0; solution is any real number {ℝ} ) and contradiction (no solution is true {∅} ) equations.
Define ratio, rate and proportion.
! I dunno about this. More usually seems to need simplification with LCD! Solve proportion equations by starting with cross multiplication then the four steps.

Solve linear equations in one variable

Success

This is a rubicon. It's the precise moment school maths left me behind, and I had no idea what was going on from here to the GCSE exams. Doing this and enjoying it brought a tear to my eye.

Fourish step process:

If it's a word or application problem, find what is being asked and assign the unknown quantity a variable.
If it's a proportion equation, one fraction each side, cross multiply.

Simplify both sides so far as possible

Remove brackets by distributive property of multiplication
When we encounter an equation with fractions and parentheses are involved, we want to remove parentheses before we clear our fractions
Q: decimals clear before or after brackets? A: before but don't multiply up the bracket contents.
Collect like terms

Isolate the variable term(s) on one side, usually left, using addition/subtraction (addition property of equality, additive inverse property)
Isolate the variable using multiplication and division
Check the answer by substituting the variable in the original equation. This matters it's where you may find div by zero etc. contradictions.

Example

Textbook Notes

Direct relevant blocks from Zaigralin - Basic Algebra with Applications 6e - 2018 - Open Textbook Library.pdf

1 - Properties of equations

A number is called a solution for the equation involving one variable if substituting this number for the variable makes the equation true. The set of all solutions for the equation is called its solution set. To solve the equation means to find its solution set.

Equations are equivalent if they have the same solutions sets.

2 - Solving linear equations

A linear equation is an equation in which each term is either a numerical constant or the product of a numerical constant and the first power of a single variable.

If the equation looks like 0 = 4 or something equally wrong, then it is false for all x. There are no solutions: the set of solutions is empty, or ∅.

If the equation looks like 0 = 0, then it is true for all x. Every real number is a solution: the set of solutions is the set of all reals, or ℝ.

3 - Linear formulas

While the word formula has no traditional formal definition, this will be our word of choice for equations in more than one variable.

We say that a formula involving a variable y is solved for y if it is written in the form y = expression where one side of the equation is just y, and the expression on the other side does not contain y at all.