Algebra 1 notes

An Equation is a mathematical statement showing that two expressions are equal. A Variable is a letter or symbol representing a value that can change. A Constant is a value that remains fixed. A Coefficient is a number multiplied by a variable in an algebraic expression. Terms are parts of an expression or equation, separated by + or - signs. Like Terms are terms with the same variables raised to the same powers. Combining Like Terms simplifies an expression by grouping like terms together. An Inequality is a mathematical sentence comparing two expressions using symbols like <, >, ≤, or ≥. The Domain is the set of all possible input values (x-values) for which a function is defined. The Range is the set of all possible output values (y-values) of a function. An Expression is a relationship or expression involving one or more variables. A Solution is a value that, when substituted for a variable, makes an equation true. Solving is the process of finding the value(s) that satisfy an equation. A Base is a number or expression raised to a power. An Exponent is a quantity representing the power to which a number or expression is raised. An Algebraic Expression is a mathematical phrase involving numbers, variables, and operations. A Comparison is a statement indicating how two numbers or expressions relate. Prime Factorization is the representation of a number as a product of its prime factors. A Linear Graph is a straight-line graph representing a linear equation. A Linear Function is a function whose graph is a straight line. Slope-Intercept Form is a function written as f(x) = mx + b, where m is the slope and b is the y-intercept. Slope is the steepness of a line, calculated as the change in y divided by the change in x. The Y-Intercept is the point where a line crosses the y-axis, represented as (

Here's how to solve each of the example equations:

1. Linear Equation: 2x + 3 = 7

   • Subtract 3 from both sides: 2x = 4

   • Divide by 2: x = 2

2. Quadratic Equation: x^2 - 4x + 4 = 0

   • Factor the quadratic: (x - 2)(x - 2) = 0

   • Set each factor equal to zero: x - 2 = 0

   • Solve for x: x = 2

3. Equation with Fractions: \frac{x}{2} + \frac{1}{3} = 1

   • Find a common denominator (6): \frac{3x}{6} + \frac{2}{6} = \frac{6}{6}

   • Multiply all terms by 6: 3x + 2 = 6

   • Subtract 2 from both sides: 3x = 4

   • Divide by 3: x = \frac{4}{3}

4. Equation with Parentheses: 3(x - 2) = 9

   • Distribute the 3: 3x - 6 = 9

   • Add 6 to both sides: 3x = 15

   • Divide by 3: x = 5

5. Equation with Exponents: x^3 = 8

   • Take the cube root of both sides: x = \sqrt[3]{8}

   • Simplify: x = 2

6. Equation with Absolute Value: |x - 1| = 2

   • Set up two equations: x - 1 = 2 and x - 1 = -2

   • Solve for x in both equations:

     • x - 1 = 2 \Rightarrow x = 3

     • x - 1 = -2 \Rightarrow x = -1

7. Equation with Two Variables: 2x + y = 5

   • This is a linear equation in two variables. To solve for one variable in terms of the other, isolate one variable.

   • Solve for y: y = 5 - 2x

   • Solutions are pairs of (x, y) values that satisfy the equation. For example, if x = 1

A linear function is a function whose graph is a straight line. It can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept. Here are a few examples of how to solve a few problems involving linear functions:

1. Finding the equation of a line given two points: Suppose we have two points (x1, y1) and (x2, y2). Let's say the points are (1, 3) and (2, 5).

   • First, find the slope (m) using the formula: m = \frac{y2 - y1}{x2 - x1}. Plugging in our values, we get: m = \frac{5 - 3}{2 - 1} = \frac{2}{1} = 2.

   • Next, use the point-slope form of a line, which is y - y1 = m(x - x1). Plug in one of the points (for example, (1, 3)) and the slope we found: y - 3 = 2(x - 1).

   • Simplify this equation to get the slope-intercept form: y - 3 = 2x - 2. y = 2x + 1. So, the equation of the line is f(x) = 2x + 1.

2. Finding the y-intercept and slope from an equation: Given the equation f(x) = -3x + 4, the slope is the coefficient of x, which is -3, and the y-intercept is the constant term, which is 4. Therefore, the slope is -3 and the y-intercept is $$(0, 4)$