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 $(</p><p>Here's how to solve each of the example equations:</p><ol><li><p>Linear Equation:$2x + 3 = 7$</p><ul><li><p>Subtract 3 from both sides:$2x = 4$</p></li><li><p>Divide by 2:$x = 2$</p></li></ul></li><li><p>Quadratic Equation:$x^2 - 4x + 4 = 0$</p><ul><li><p>Factor the quadratic:$(x - 2)(x - 2) = 0$</p></li><li><p>Set each factor equal to zero:$x - 2 = 0$</p></li><li><p>Solve for x:$x = 2$</p></li></ul></li><li><p>Equation with Fractions:$\frac{x}{2} + \frac{1}{3} = 1$</p><ul><li><p>Find a common denominator (6):$\frac{3x}{6} + \frac{2}{6} = \frac{6}{6}$</p></li><li><p>Multiply all terms by 6:$3x + 2 = 6$</p></li><li><p>Subtract 2 from both sides:$3x = 4$</p></li><li><p>Divide by 3:$x = \frac{4}{3}$</p></li></ul></li><li><p>Equation with Parentheses:$3(x - 2) = 9$</p><ul><li><p>Distribute the 3:$3x - 6 = 9$</p></li><li><p>Add 6 to both sides:$3x = 15$</p></li><li><p>Divide by 3:$x = 5$</p></li></ul></li><li><p>Equation with Exponents:$x^3 = 8$</p><ul><li><p>Take the cube root of both sides:$x = \sqrt[3]{8}$</p></li><li><p>Simplify:$x = 2$</p></li></ul></li><li><p>Equation with Absolute Value:$|x - 1| = 2$</p><ul><li><p>Set up two equations:$x - 1 = 2$and$x - 1 = -2$</p></li><li><p>Solve for x in both equations:</p><ul><li><p>$x - 1 = 2 \Rightarrow x = 3$</p></li><li><p>$x - 1 = -2 \Rightarrow x = -1$</p></li></ul></li></ul></li><li><p>Equation with Two Variables:$2x + y = 5$</p><ul><li><p>This is a linear equation in two variables. To solve for one variable in terms of the other, isolate one variable.</p></li><li><p>Solve for y:$y = 5 - 2x$</p></li><li><p>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 $(x<em>1, y</em>1)$ and $(x<em>2, y</em>2)$. Let's say the points are $(1, 3)$ and $(2, 5)$.

   • First, find the slope (m) using the formula: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$. 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 $y - y<em>1 = m(x - x</em>1)$. 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)$