# A.4ef and *A.4de - Systems of Linear Equations

A.4    The student will solve multistep linear and quadratic equations in two variables, including

e)  solving systems of two linear equations in two variables algebraically and graphically; and

f)  solving real-world problems involving equations and systems of equations.

Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.

A.4  The student will solve

d)  systems of two linear equations in two variables algebraically and graphically;

e)  practical problems involving equations and systems of equations.

Bloom's Level:  Analyze

### BIG IDEAS

• I can find how far and how fast a bus travels, the number of fruits that can be purchased, and how long it takes to drain a swimming pool.
• I will be able to write symbolic representations of the way numbers behave and will know that in order to maintain equality, an operation performed on one side must also be performed on the other side.

### UNDERSTANDING THE STANDARD

• A solution to an equation is the value or set of values that can be substituted to make the equation true.
• The solution of an equation in one variable can be found by graphing the expression on each side of the equation separately and finding the x-coordinate of the point of intersection.
• Real-world problems can be interpreted, represented, and solved using linear and quadratic equations.
• The process of solving linear and quadratic equations can be modeled in a variety of ways, using concrete, pictorial, and symbolic representations.
• Properties of real numbers and properties of equality can be used to justify equation solutions and expression simplification.
• The zeros or the x-intercepts of the quadratic function are the real root(s) or solution(s) of the quadratic equation that is formed by setting the given quadratic expression equal to zero.
• A system of linear equations with exactly one solution is characterized by the graphs of two lines whose intersection is a single point, and the coordinates of this point satisfy both equations.
• A system of two linear equations with no solution is characterized by the graphs of two lines that are parallel.
• A system of two linear equations having infinite solutions is characterized by two graphs that coincide (the graphs will appear to be the graph of one line), and the coordinates of all points on  the line satisfy both equations.
• Systems of two linear equations can be used to model two real-world conditions that must be satisfied simultaneously.
• Equations and systems of equations can be used as mathematical models for real-world situations.
• Set builder notation may be used to represent solution sets of equations.

·  A solution to an equation is the value or set of values that can be substituted to make the equation true.

·  Each point on the graph of a linear or quadratic equation in two variables is a solution of the equation.

·  Practical problems may be interpreted, represented, and solved using linear and quadratic equations.

·  The process of solving linear and quadratic equations can be modeled in a variety of ways, using concrete, pictorial, and symbolic representations.

·  Properties of real numbers and properties of equality are applied to solve equations.

·  Properties of Real Numbers:

­  Associative Property of Multiplication

­  Commutative Property of Multiplication

­  Identity Property of Multiplication (Multiplicative Identity)

­  Inverse Property of Multiplication (Multiplicative Inverse)

­  Distributive Property

·  Properties of Equality:

­  Multiplicative Property of Zero

­  Zero Product Property

­  Reflexive Property

­  Symmetric Property

­  Transitive Property of Equality

­  Subtraction Property of Equality

­  Multiplication Property of Equality

­  Division Property of Equality

­  Substitution

·  A system of linear equations with exactly one solution is characterized by the graphs of two lines whose intersection is a single point, and the coordinates of this point satisfy both equations.

### ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

A.4e2  Given a system of two linear equations in two variables that has a unique solution, solve the system graphically by identifying the point of intersection.

·  A.4dGiven a system of two linear equations in two variables that has a unique solution, solve the system graphically by identifying the point of intersection.

A.4e1  Given a system of two linear equations in two variables that has a unique solution, solve the system by substitution of elimination to find the ordered pair which satisfies both equations.

·  A.4dGiven a system of two linear equations in two variables that has a unique solution, solve the system by substitution or elimination to identify the ordered pair which satisfies both equations.

·  A.4dSolve and confirm algebraic solutions to a system of two linear equations using a graphing utility.

A.4e3  Determine whether a system of two linear equations has one solutions, no solution, or infinite solutions.

·  A.4dDetermine whether a system of two linear equations has one, an infinite number, or no solutions.

A.4e4  Write a system of two linear equations that models a real-world situation.

·  A.4eWrite a system of two linear equations that models a practical situation.

A.4f1  Interpret and determine the reasonableness of the algebraic or graphical solution of a system of two linear equations that models a real-world situation.

·  A.4eInterpret and determine the reasonableness of the algebraic or graphical solution of a system of two linear equations that models a practical situation.

·  A.4eSolve practical problems involving equations and systems of equations.

### KEY VOCABULARY

literal equation (formula), expression, equation, properties of real numbers, properties of equality, quadratic equation, quadratic function, quadratic expression, roots, zeros, linear equation, system, substitution, elimination, ordered pair, coordinate, intersection, intercepts, solution, infinite, parallel, coincide, simultaneous

Updated: Oct 27, 2017