#### Math - 2017-18

# 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.4The student willsolved) systems of two linear equations in two variables algebraically and graphically;

e) practical problems involving equations and systems of equations.

**Bloom's Level: ** Analyze

*Adopted: 2009*

### 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.

2016 VDOE Curriculum Framework - AI.4 Understanding

· 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 Addition

Associative Property of Multiplication

Commutative Property of Addition

Commutative Property of Multiplication

Identity Property of Addition (Additive Identity)

Identity Property of Multiplication (Multiplicative Identity)

Inverse Property of Addition (Additive Inverse) 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

Addition 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.4e_{2} 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.4d_{2 }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.4e_{1} 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.4d_{1 }Given 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.4d_{3 }**Solve**
and **confirm** algebraic solutions to a
system of two linear equations using a graphing utility.

A.4e_{3} **Determine**
whether a system of two linear equations has one solutions, no solution, or
infinite solutions.

·
A.4d_{4 }**Determine**
whether a system of two linear equations has one, an infinite number, or no
solutions.

A.4e_{4} **Write**
a system of two linear equations that models a real-world situation.

·
A.4e_{1 }**Write**
a system of two linear equations that models a practical situation.

A.4f_{1} **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.4e_{2 }**Interpret**
and **determine** the reasonableness of
the algebraic or graphical solution of a system of two linear equations that
models a practical situation.

·
A.4e_{3 }**Solve**
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*