#### Math - 2019-20

# 7.12 - Linear Equations

7.12The student willsolvetwo-step linear equations in one variable, including practical problems that require the solution of a two-step linear equation in one variable.

*Adopted: 2016*

### BIG IDEAS

- I can find how far and how fast a bus travels, the number of fruits that can be purchased, how long it takes to
drain a swimming pool, how many hours I can afford to rent
roller blades when there is a cost for skate rental plus a per hour charge.

- I can translate real-life situations into equations to find unknown values and I will know that to maintain equality, an operation performed on one side of an equation must also be performed on the other side.

### UNDERSTANDING THE STANDARD

· An equation is a mathematical sentence that states that two expressions are equal.

· The solution to an equation is the value(s) that make it a true statement. Many equations have one solution and can be represented as a point on a number line.

· A variety of concrete materials such as colored chips, algebra tiles, or weights on a balance scale may be used to model solving equations in one variable.

· The inverse operation for addition is subtraction, and the inverse operation for multiplication is division.

·
A two-step
equation may include, but not be limited to equations such as the following:

2*x* + = -5; -25 = 7.2*x*
+ 1; = 4; *x* – 2 = 10.

·
An
expression is a representation of quantity. It may contain numbers, variables,
and/or operation symbols. It does not have an “equal sign (=)” (e.g.,, 5*x*, 140 - 38.2, 18 ∙
21, 5 + *x*).

·
An
expression that contains a variable is a variable expression. A variable
expression is like a phrase: as a phrase does not have a verb, so an expression
does not have an “equal sign (=).”

An expression cannot be solved.

·
A verbal
expression can be represented by a variable expression. Numbers are used when
they are known; variables are used when the numbers are unknown. For example, the verbal expression “a number
multiplied by 5” could be represented by “*n*
∙ 5” or “5*n*”.

·
An algebraic
expression is a variable expression that contains at least one variable (e.g.,
2*x* – 3).

·
A verbal
sentence is a complete word statement (e.g., “The sum of twice a number and two
is fifteen.” could be represented by “2*n*
+ 2 = 15”).

·
An algebraic
equation is a mathematical statement that says that two expressions are equal

(e.g., 2*x *– 8 = 7).

·
Properties
of real numbers and properties of equality can be applied when solving
equations, and justifying solutions. Students should use the following
properties, where appropriate, to further develop flexibility and fluency in
problem solving (limitations may exist for the values of *a*, *b*, or *c* in this standard):

- Commutative property of addition: .

- Commutative property of multiplication: .

- Subtraction and division are not commutative.

- The additive identity is zero (0) because any number added to zero is the number. The multiplicative identity is one (1) because any number multiplied by one is the number. There are no identity elements for subtraction and division.

- Identity property of addition (additive identity property): .

- Identity property of multiplication (multiplicative identity property): .

- Inverses are numbers that combine with other
numbers and result in identity elements

(e.g., 5 + (–5) = 0; · 5 = 1).

- Inverse property of addition (additive inverse property): .

- Inverse property of multiplication (multiplicative inverse property): .

- Zero has no multiplicative inverse.

- Multiplicative property of zero: .

- Division by zero is not a possible mathematical operation. It is undefined.

- Substitution property: If , then *b* can be
substituted for *a* in any expression,
equation, or inequality.

- Addition property of equality: If , then .

- Subtraction property of equality: If then .

- Multiplication property of equality: If then .

- Division property of equality: If then .### ESSENTIALS

- When solving an
equation, why is it important to perform identical operations on each side of
the equal sign?

An operation that is performed on one side of an equation must be performed on the other side to maintain equality.

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

·
7.12_{1} **Represent**
and **solve** two-step linear equations
in one variable using a variety of concrete materials and pictorial
representations.

·
7.12_{2} **Apply**
properties of real numbers and properties of equality to **solve **two-step linear
equations in one variable. Coefficients and numeric terms will be rational.

·
7.12_{3} **Confirm**
algebraic solutions to linear equations in one variable.

·
7.12_{6} **Solve**
practical problems that require the solution of a two-step linear equation.

### KEY VOCABULARY

linear equation, variable, solution, inverse operation, algebraic
sentence, balanced, solve, represent

*Updated: Jul 30, 2019*