Reasoning with Equations and Inequalities

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Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.
Solve quadratic equations in one variable.
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x2 + y2 = 3.
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Examples: Rearrange Ohms law V = IR to highlight resistance R; Rearrange area of a circle formula A = ?r2 to highlight the radius r.
Using algebraic properties and the properties of real numbers, justify the steps of a simple, one-solution equation. Students should justify their own steps, or if given two or more steps of an equation, explain the progression from one step to the next using properties.
Using graphs, tables, or successive approximations, show that the solution to the equation f(x) = g(x) is the x-value where the y-values of f(x) and g(x) are the same.
Graph the solution set to a linear inequality in two variables.
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. For example, given ax + 3 = 7, solve for x.
Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p)2 = q that has the same solutions. Derive the quadratic formula from ax2 + bx + c = 0.
Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions).
Show and explain why the elimination method works to solve a system of two-variable equations.
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Represent a system of linear equations as a single matrix equation in a vector variable
Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 3 or greater).

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