Math 101: Solving Linear Equations

• Math 101 covers sections 2.1-2.4 and 3.1.

Section 2.1: Linear Equations

• A linear equation in one variable can be written as ax + b = 0, where a and b represent real numbers, and a ≠ 0.
• Solving a linear equation involves isolating the variable on one side.
• Isolation is achieved by employing the addition, subtraction, multiplication, and division properties of equality.
• Equality properties allow performing the same operation on both equation sides.
• Manipulating the equation aims to achieve the form x = constant.
• Example: 2x + 3 = 7 can be solved by subtracting 3 from both sides (2x = 4), then dividing by 2 (x = 2).
• Equations sometimes require simplification using the distributive propriety.
• No solution can occur which leads to a contradiction, such as 0 = 1.
• Infinite solutions are possible which leads to an identity, such as 0 = 0.

Section 2.2: Applications of Linear Equations

• Focus is on translating word problems into mathematical equations.
• Requires identifying the unknown quantity and assigning a variable.
• Translation of information transforms word problems into mathematical expressions using variables.
• Equality (=) is often indicated by keywords: "is," "equals," "was," and "results in".
• Addition (+) is usually indicated by "more than" or "increased by". Subtraction (-) is indicated by "less than" or "decreased by".
• "Of" indicates multiplication.
• Geometry formulas are frequently utilized, familiarity with area, perimeter, and volume formulas is important.
• Distance problems often use the formula distance = rate × time (d = rt).
• Mixture problems combine two or more quantities with different concentrations or values.

Section 2.3: Formulas and Problem Solving

• Relationships between two or more variables are expressed through formulas.
• Solving a formula for a variable involves isolating that variable on one side.
• The same properties of equality that are applied to solve linear equations are also used here.
• Given A = lw, solving for w requires dividing both sides by l resulting in w = A/l.
• Interest formula: I = PRT (I = interest, P = principal, R = rate, T = time).
• Percent formula: amount = percent * base.
• Temperature formula: F = (9/5)C + 32 relates degrees Fahrenheit to degrees Celsius.

Section 2.4: Linear Inequalities

• A linear inequality is a statement comparing two expressions using inequality symbols.
• Common symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).
• Solving linear inequalities is similar to solving linear equations with one crucial difference.
• Multiplying or dividing by a negative number reverses inequality direction.
• Example: -2x < 6. Dividing by -2 gives x > -3 (inequality sign is flipped).
• Solution sets can be represented graphically on a number line.
• An open circle is used for < and >, indicating the endpoint isn't in the solution.
• A closed circle is used for ≤ and ≥, indicating the endpoint is included.
• Solution sets are expressed in interval notation.
• Example: x > 3 is written as (3, ∞).
• Example: x ≤ 5 is written as (-∞, 5].
• Inequalities can model and solve real-world problems.

Section 3.1: Graphing Equations

• The Cartesian coordinate system is used to represent points in a two-dimensional plane.
• It has two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).
• The origin is where the axes intersect and is represented by (0, 0).
• Points in the plane are identified by ordered pairs (x, y).
• The x-coordinate represents the horizontal distance from the origin, and the y-coordinate represents the vertical distance.
• Graphing an equation involves plotting all ordered pairs (x, y) that satisfy it.
• For linear equations, this typically results in a straight line.
• To graph a linear equation, find at least two points that satisfy the equation and then draw a line through them.
• The x-intercept is the point where the line crosses the x-axis (y = 0).
• The y-intercept is the point where the line crosses the y-axis (x = 0).
• Intercepts are useful for graphing linear equations.