Topic: Reasoning with Equations and Inequalities |
Common Core Mathematics 9-12 |
| Cluster Understand solving equations as a process of reasoning and explain the reasoning |
| | Grade 9-12 |
| | | A.REI.1. | Explain each step in solving a simple equation as following from the
equality of numbers asserted at the previous step, starting from the
assumption that the original equation has a solution. Construct a
viable argument to justify a solution method. |
| | | A.REI.2. | Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise. |
| Cluster Solve equations and inequalities in one variable |
| | Grade 9-12 |
| | | A.REI.3. | Solve linear equations and inequalities in one variable, including
equations with coefficients represented by letters. |
| | | A.REI.4. | Solve quadratic equations in one variable. |
| | | A.REI.4.a. | 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 this form. |
| | | A.REI.4.b. | Solve quadratic equations by inspection (e.g., for x^{2} = 49), taking
square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. |
| Cluster Solve systems of equations |
| | Grade 9-12 |
| | | A.REI.5. | Prove that, given a system of two equations in two variables, replacing
one equation by the sum of that equation and a multiple of the other
produces a system with the same solutions. |
| | | A.REI.6. | Solve systems of linear equations exactly and approximately (e.g., with
graphs), focusing on pairs of linear equations in two variables. |
| | | A.REI.7. | 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 x^{2} + y^{2} = 3. |
| | | A.REI.8. | (+) Represent a system of linear equations as a single matrix equation
in a vector variable. |
| | | A.REI.9. | (+) 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). |
| Cluster Represent and solve equations and inequalities graphically |
| | Grade 9-12 |
| | | A.REI.10. | Understand that the graph of an equation in two variables is the set of
all its solutions plotted in the coordinate plane, often forming a curve
(which could be a line). |
| | | A.REI.11. | Explain why the x-coordinates of the points where the graphs of
the equations y = f(x) and y = g(x) intersect are the solutions of the
equation f(x) = g(x); find the solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find
successive approximations. Include cases where f(x) and/or g(x)
are linear, polynomial, rational, absolute value, exponential, and
logarithmic functions. |
| | | A.REI.12. | Graph the solutions to a linear inequality in two variables as a halfplane
(excluding the boundary in the case of a strict inequality), and
graph the solution set to a system of linear inequalities in two variables
as the intersection of the corresponding half-planes. |