Contenido principal

### Math

## Common Core Math

# High School: Algebra: Reasoning with Equations and Inequalities

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.

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

- Equation that has a specific extraneous solution
- Equations with one rational expression (advanced)
- Equations with rational expressions
- Equations with rational expressions (example 2)
- Extraneous solutions
- Extraneous solutions of equations
- Extraneous solutions of radical equations
- Find inverses of rational functions
- Finding inverses of rational functions
- Intro to solving square-root equations
- Intro to square-root equations & extraneous solutions
- Rational equations
- Rational equations (advanced)
- Rational equations intro
- Rational equations intro
- Solving cube-root equations
- Solving square-root equations
- Solving square-root equations (basic)
- Solving square-root equations: no solution
- Solving square-root equations: one solution
- Solving square-root equations: two solutions
- Square-root equations
- Square-root equations intro
- Square-root equations intro

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

- A compound inequality with no solution
- Compound inequalities
- Compound inequalities examples
- Compound inequalities review
- Compound inequalities: AND
- Compound inequalities: OR
- Double inequalities
- Linear equations with unknown coefficients
- Linear equations with unknown coefficients
- Multi-step linear inequalities

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.

- Completing the square
- Completing the square
- Completing the square (intermediate)
- Completing the square (intro)
- Completing the square review
- Proof of the quadratic formula
- Quadratic formula proof review
- Solving quadratics by completing the square
- Solving quadratics by completing the square: no solution
- Worked example: completing the square (leading coefficient ≠ 1)
- Worked example: Rewriting & solving equations by completing the square
- Worked example: Rewriting expressions by completing the square

Solve quadratic equations by inspection, 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.

- Completing the square
- Completing the square (intermediate)
- Completing the square (intro)
- Completing the square review
- Discriminant review
- Number of solutions of quadratic equations
- Proof of the quadratic formula
- Quadratic equations word problem: box dimensions
- Quadratic equations word problem: triangle dimensions
- Quadratic formula
- Quadratic formula proof review
- Quadratic formula review
- Quadratic word problem: ball
- Quadratic word problems (standard form)
- Quadratics by factoring
- Quadratics by factoring (intro)
- Quadratics by taking square roots
- Quadratics by taking square roots (intro)
- Quadratics by taking square roots: strategy
- Quadratics by taking square roots: strategy
- Quadratics by taking square roots: with steps
- Solve by completing the square: Integer solutions
- Solve by completing the square: Non-integer solutions
- Solve equations by completing the square
- Solve equations using structure
- Solve quadratic equations: complex solutions
- Solving quadratic equations: complex roots
- Solving quadratics by completing the square
- Solving quadratics by completing the square: no solution
- Solving quadratics by factoring
- Solving quadratics by factoring
- Solving quadratics by factoring review
- Solving quadratics by factoring: leading coefficient ≠ 1
- Solving quadratics by taking square roots
- Solving quadratics by taking square roots
- Solving quadratics by taking square roots examples
- Solving quadratics by taking square roots: challenge
- Solving quadratics by taking square roots: with steps
- Solving quadratics using structure
- Solving simple quadratics review
- Strategy in solving quadratic equations
- Strategy in solving quadratics
- The quadratic formula
- Understanding the quadratic formula
- Using the quadratic formula
- Using the quadratic formula: number of solutions
- Worked example: completing the square (leading coefficient ≠ 1)
- Worked example: quadratic formula
- Worked example: quadratic formula (example 2)
- Worked example: quadratic formula (negative coefficients)
- Worked example: Rewriting & solving equations by completing the square
- Worked example: Rewriting expressions by completing the square
- Zero product property
- Zero product property

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.

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

- Age word problem: Arman & Diya
- Age word problem: Ben & William
- Age word problem: Imran
- Age word problems
- Combining equations
- Creating systems in context
- Elimination method review (systems of linear equations)
- Elimination strategies
- Elimination strategies
- Linear systems of equations capstone
- Reasoning with systems of equations
- Setting up a system of equations from context example (pet weights)
- Setting up a system of linear equations example (weight and price)
- Solutions of systems of equations
- Substitution method review (systems of equations)
- System of equations word problem: infinite solutions
- System of equations word problem: no solution
- System of equations word problem: walk & ride
- Systems of equations with elimination
- Systems of equations with elimination (and manipulation)
- Systems of equations with elimination challenge
- Systems of equations with elimination: apples and oranges
- Systems of equations with elimination: coffee and croissants
- Systems of equations with elimination: King's cupcakes
- Systems of equations with elimination: potato chips
- Systems of equations with elimination: TV & DVD
- Systems of equations with elimination: x-4y=-18 & -x+3y=11
- Systems of equations with graphing
- Systems of equations with graphing: exact & approximate solutions
- Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
- Systems of equations with substitution
- Systems of equations with substitution: -3x-4y=-2 & y=2x-5
- Systems of equations with substitution: coins
- Systems of equations with substitution: potato chips
- Systems of equations word problems
- Systems of equations word problems (with zero and infinite solutions)
- Systems of equations: trolls, tolls (1 of 2)
- Systems of equations: trolls, tolls (2 of 2)
- Testing a solution to a system of equations

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

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).

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).

- Complete solutions to 2-variable equations
- Completing solutions to 2-variable equations
- How many solutions does a system of linear equations have if there are at least two?
- Intercepts from a graph
- Intercepts from an equation
- Intro to intercepts
- Intro to linear equation standard form
- Intro to point-slope form
- Intro to slope-intercept form
- Intro to slope-intercept form
- Number of solutions to a system of equations
- Number of solutions to a system of equations algebraically
- Number of solutions to a system of equations algebraically
- Number of solutions to a system of equations graphically
- Number of solutions to a system of equations graphically
- Number of solutions to system of equations review
- Solutions to 2-variable equations
- Solutions to 2-variable equations
- Solutions to systems of equations: consistent vs. inconsistent
- Solutions to systems of equations: dependent vs. independent
- Systems of equations number of solutions: fruit prices (1 of 2)
- Systems of equations number of solutions: fruit prices (2 of 2)
- Two-variable linear equations intro
- Worked example: solutions to 2-variable equations

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.

- How many solutions does a system of linear equations have if there are at least two?
- Interpret equations graphically
- Interpreting equations graphically
- Interpreting equations graphically (example 2)
- Number of solutions to a system of equations
- Number of solutions to a system of equations algebraically
- Number of solutions to a system of equations algebraically
- Number of solutions to a system of equations graphically
- Number of solutions to a system of equations graphically
- Number of solutions to system of equations review
- Solutions of systems of equations
- Solutions to systems of equations: consistent vs. inconsistent
- Solutions to systems of equations: dependent vs. independent
- Solve equations graphically
- Solving equations by graphing
- Solving equations by graphing: graphing calculator
- Solving equations by graphing: intro
- Solving equations by graphing: word problems
- Solving equations graphically
- Solving equations graphically (1 of 2)
- Solving equations graphically (2 of 2)
- Solving equations graphically: graphing calculator
- Solving equations graphically: intro
- Solving equations graphically: word problems
- Systems of equations number of solutions: fruit prices (1 of 2)
- Systems of equations number of solutions: fruit prices (2 of 2)
- Systems of equations with graphing
- Systems of equations with graphing: exact & approximate solutions
- Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
- Systems of equations: trolls, tolls (1 of 2)
- Systems of equations: trolls, tolls (2 of 2)
- Testing a solution to a system of equations

Graph the solutions to a linear inequality in two variables as a half-plane (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.

- Graphing inequalities (x-y plane) review
- Graphing systems of inequalities
- Graphing two-variable inequalities
- Graphs of inequalities
- Interpret points relative to a system
- Intro to graphing systems of inequalities
- Intro to graphing two-variable inequalities
- Systems of inequalities graphs
- Two-variable inequalities from their graphs
- Two-variable inequalities from their graphs