algebra silvong linears 9/3
Page 1: Overview of solving linear equations and equation types
Goal of solving equations: maintain the same standard process across different types of equations. The process does not change; what changes is the kind of solution you end up with.
General solving strategy (unchanged):
Step 1: Simplify, if possible.
Step 2: Move variable terms to one side and constant terms to the other side.
Step 3: Use multiplication or division to isolate the target variable so that its coefficient becomes 1.
Types of equations we’ll encounter when solving for one variable (x):
Conditional equations: have exactly one numerical solution.
Contradictions: have no solution (no real number satisfies the equation).
Identities: have infinitely many solutions; every real number satisfies the equation.
Key notations:
A single-solution equation is called conditional.
An equation with no solution is called a contradiction.
An equation true for all real numbers is called an identity, and its solution set is all real numbers, denoted by the symbol for all real numbers \mathbb{R}.
Important reminders:
Zero is a real number and can be a solution (a conditional equation can have x = 0 as its solution).
No solution is not the same as the number zero; a contradiction does not evaluate to a number like 0, it has no solution.
When an equation has all real numbers as solutions, we call it an identity.
Page 2: Definitions and implications of the three solution types
Conditional equation:
Has exactly one numerical solution, e.g., x = 0 or x = \frac{3}{4}.
Example (illustrative): if after solving you obtain a single value for x, that is a conditional equation.
Contradiction (no solution):
There is no real number that satisfies the equation.
Recognize it when, after simplification, you get a false statement like 5 = -7 or two equal variable terms with different constants (e.g., 2x + 7 = 2x + 9 would simplify to 7 = 9).
Identity (all real numbers):
Every real number satisfies the equation.
Recognize it when, after simplification, you obtain the same expression on both sides (e.g., 3x+2 = 2+3x).
The solution set is \mathbb{R} (all real numbers).
Page 3: How to recognize contradictions vs identities while solving
If, after simplification, both sides have the same variable terms but different constants, you typically have a contradiction (no solution).
Example clue: left side has 2x+7, right side has 2x+9. The variable terms match, but the constants do not.
In this case, you can stop early and conclude there is no solution (a contradiction).
If simplifying yields identical expressions on both sides (same variable terms and same constants), you have an identity (all real numbers).
Example clue: 3x+2 = 2+3x; the sides match term-for-term after rearrangement, so any real number is a solution.
If simplifying yields a true numerical statement after eliminating the variables (e.g., 7 = 7), then the equation is an identity and the solution set is \mathbb{R}.
If none of the above early recognitions apply, continue solving with the standard steps to determine whether you have a conditional, contradiction, or identity at the end.
Page 4: Contradiction example (no solution)
Example: solving 2x + 7 = 2x + 9
After simplifying, the 2x terms cancel, leaving 7 = 9, which is false.
This indicates a contradiction and thus no solution.
Important: do not reinterpret this as a solution of 0; there is no solution at all.
Another way a contradiction can appear is when, during solving, you arrive at a false numerical statement like 5 = -7 (with no variables remaining).
Takeaway: identical variable terms with non-matching constants on each side signal a contradiction and no solution.
Page 5: Identity example (all real numbers)
Example: solving 3x + 2 = 2 + 3x
After recognizing the same x-term on both sides and the same constant on both sides, the equation is an identity.
Any real number is a solution; the solution set is \mathbb{R}.
Quick rule: if simplifying yields identical expressions on both sides, mark it as an identity and state all real numbers as the solution set.
Page 6: Conditional example (one numerical solution)
Example to illustrate a single-solution case: 3x = 0
Solve by isolating x: divide both sides by 3 to get x = 0.
This is a conditional equation with the unique solution x = 0.
Summary: after solving, you should also label the type of equation (conditional in this case) and state the solution set, e.g., {0}.
Page 7: A more involved conditional example and the idea of fractions
Another conditional example that yields a finite solution with a fraction might arise, e.g., solving for x when dividing by a nonzero coefficient produces a reduced fraction.
Suppose after solving you arrive at a reduced fraction, e.g., x = \dfrac{9}{4}. It is still a conditional equation because there is one real-number solution, just not an integer.
Key point: when fractions appear, reduce them if possible to their lowest terms (e.g., reduce \frac{27}{12} to \frac{9}{4}).
Page 8: Solving equations with two variables for one variable (e.g., solve for y)
Concept: When you have a single equation in two variables (for example, ax + by = c), you can solve for one variable in terms of the other.
General method to solve for y from ax + by = c:
Move the term involving x to the other side: by = c - ax.
Divide by the coefficient of y (b): y = \dfrac{c - ax}{b}.
Example (illustrative): starting with 6y = 3x + 12, solve for y:
Divide both sides by 6: y = \dfrac{3x + 12}{6} = \dfrac{1}{2}x + 2.
Notes on two-variable problems:
The role of the equation is to relate x and y; the solution is a line of pairs (x, y) that satisfy the equation.
Fractions can appear; reduce when possible.
This process mirrors the single-variable approach, but you are explicitly solving for a different variable.
Page 9: Summary of practice strategies and notation
Always start by simplifying if possible.
Then move variable terms to one side and constants to the other.
Determine the type of equation after simplification and, if possible, early stop when you can identify the type (especially in cases of a clear contradiction or identity).
If you must continue solving, apply the same steps you used for one-variable equations, keeping track of whether the final result is a conditional (one solution), a contradiction (no solution), or an identity (all real numbers).
Key terminology recap:
Conditional equation: exactly one numerical solution; notation for the solution set is a singleton, e.g., {0}.
Contradiction: no solution; cannot satisfy the equation by any real number.
Identity: all real numbers solve the equation; solution set is \mathbb{R}.
Quick reminder on solution sets:
Conditional: {x0} for the single solution x0.
Identity: \mathbb{R} (all real numbers).
Real-number notation: all real numbers is often written as \mathbb{R} in math notation.
Page 10: Quick reference to the common case distinctions
If after simplification you have the same variable terms on both sides but different constants: contradiction (no solution).
If after simplification you have identical expressions on both sides: identity (all real numbers).
If you end up with a single numeric solution for x: conditional (solution set is a singleton).
In two-variable problems, solving for one variable (e.g., y) yields an expression in terms of the other variable; e.g., y = \dfrac{c - ax}{b}, which may include fractions.
(Note: ALEKS Section 1.2 assignment information mentioned in class.)