Algebra Systems of Equations

Systems of Equations

• A system of equations is a set of two or more equations with two or more variables.
• Methods for solving systems of equations:
  • Substitution method: solve one equation for one variable and substitute into the other equation.
  • Elimination method: add or subtract equations to eliminate one variable, then solve for the remaining variable.
  • Graphical method: graph the equations on a coordinate plane and find the point of intersection.

Functions and Graphs

• A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
• Key features of graphs:
  • Domain: all possible input values.
  • Range: all possible output values.
  • x-intercepts: points where the graph crosses the x-axis.
  • y-intercepts: points where the graph crosses the y-axis.
  • Asymptotes: lines that the graph approaches as x approaches positive or negative infinity.
• Graphing techniques:
  • Vertical line test: if a vertical line intersects the graph at more than one point, the relation is not a function.
  • Horizontal line test: if a horizontal line intersects the graph at more than one point, the function is not one-to-one.

Series and Sequences

• A sequence is an ordered list of numbers, denoted by {a_n}.
• A series is the sum of the terms of a sequence, denoted by ∑a_n.
• Types of sequences:
  • Arithmetic sequence: each term is obtained by adding a fixed constant to the previous term.
  • Geometric sequence: each term is obtained by multiplying the previous term by a fixed constant.
• Formulas:
  • Arithmetic series: S_n = (n/2)(a_1 + a_n)
  • Geometric series: S_n = a_1(1 - r^n) / (1 - r)

Quadratic Equations

• A quadratic equation is a polynomial equation of degree two, in the form ax^2 + bx + c = 0.
• Methods for solving quadratic equations:
  • Factoring: express the equation as a product of two binomials.
  • Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
  • Graphical method: graph the related function and find the x-intercepts.
• Key features of quadratic graphs:
  • Vertex: the minimum or maximum point of the parabola.
  • Axis of symmetry: the vertical line that passes through the vertex.

Polynomial Equations

• A polynomial equation is an equation consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• Degree of a polynomial: the highest power of the variable.
• Methods for solving polynomial equations:
  • Factoring: express the equation as a product of simpler polynomials.
  • Synthetic division: a shortcut for dividing a polynomial by a linear factor.
  • Rational root theorem: potential rational roots of a polynomial equation are of the form ±(factor of constant term) / (factor of leading coefficient).

Systems of Equations

• A system of equations consists of two or more equations with two or more variables.
• Three methods for solving systems of equations:
  • Substitution method: solve one equation for one variable and substitute into the other equation.
  • Elimination method: add or subtract equations to eliminate one variable, then solve for the remaining variable.
  • Graphical method: graph the equations on a coordinate plane and find the point of intersection.

Functions and Graphs

• A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
• Key features of graphs:
  • Domain: all possible input values.
  • Range: all possible output values.
  • X-intercepts: points where the graph crosses the x-axis.
  • Y-intercepts: points where the graph crosses the y-axis.
  • Asymptotes: lines that the graph approaches as x approaches positive or negative infinity.
• Graphing techniques:
  • Vertical line test: if a vertical line intersects the graph at more than one point, the relation is not a function.
  • Horizontal line test: if a horizontal line intersects the graph at more than one point, the function is not one-to-one.

Series and Sequences

• A sequence is an ordered list of numbers, denoted by {a_n}.
• A series is the sum of the terms of a sequence, denoted by ∑a_n.
• Types of sequences:
  • Arithmetic sequence: each term is obtained by adding a fixed constant to the previous term.
  • Geometric sequence: each term is obtained by multiplying the previous term by a fixed constant.
• Formulas:
  • Arithmetic series: S_n = (n/2)(a_1 + a_n).
  • Geometric series: S_n = a_1(1 - r^n) / (1 - r).

Quadratic Equations

• A quadratic equation is a polynomial equation of degree two, in the form ax^2 + bx + c = 0.
• Methods for solving quadratic equations:
  • Factoring: express the equation as a product of two binomials.
  • Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
  • Graphical method: graph the related function and find the x-intercepts.
• Key features of quadratic graphs:
  • Vertex: the minimum or maximum point of the parabola.
  • Axis of symmetry: the vertical line that passes through the vertex.

Polynomial Equations

• A polynomial equation is an equation consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• Degree of a polynomial: the highest power of the variable.
• Methods for solving polynomial equations:
  • Factoring: express the equation as a product of simpler polynomials.
  • Synthetic division: a shortcut for dividing a polynomial by a linear factor.
  • Rational root theorem: potential rational roots of a polynomial equation are of the form ±(factor of constant term) / (factor of leading coefficient).