# 6.18 and *6.13 - Linear Equations

6.18    The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions.

6.13  The student will solve one-step linear equations in one variable, including practical problems that require the solution of a one-step linear equation in one variable.

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

### UNDERSTANDING THE STANDARD

• A one-step linear equation is an equation that requires one operation to solve.
• A mathematical expression contains a variable or a combination of variables, numbers, and/or operation symbols and represents a mathematical relationship.  An expression cannot be solved.
• A term is a number, variable, product, or quotient in an expression of sums and/or differences. In 7x2 + 5x – 3, there are three terms, 7x2, 5x, and 3.
• A coefficient is the numerical factor in a term. For example, in the term 3xy2, 3 is the coefficient; in the term z, 1 is the coefficient.
• Positive rational solutions are limited to whole numbers and positive fractions and decimals.
• An equation is a mathematical sentence stating that two expressions are equal.
• A variable is a symbol (placeholder) used to represent an unspecified member of a set.

·  A one-step linear equation may include, but not be limited to, equations such as the following:
2x = 5;  y − 3 = −6;  x = −3;  a − (4) = 11.

·  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.

·  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.,, 5x, 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. Example, the verbal expression “a number multiplied by 5” could be represented by the variable expression “n ∙ 5” or “5n.”

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

·  A verbal sentence is a complete word statement (e.g., “The sum of a number and two is five” could be represented by “n + 2 = 5”).

·  An algebraic equation is a mathematical statement that says that two expressions are equal
(e.g., 2x = 7).

·  A term is a number, variable, product, or quotient in an expression of sums and/or differences. In 7x2 + 5x – 3, there are three terms, 7x2, 5x, and 3.

·  A coefficient is the numerical factor in a term. Example: in the term 3xy2, 3 is the coefficient; in the term z, 1 is the coefficient.

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

·  A variable is a symbol used to represent an unknown quantity.

·  The solution to an equation is a value that makes it a true statement. Many equations have one solution and are represented as a point on a number line. Solving an equation or inequality involves a process of determining which value(s) from a specified set, if any, make the equation or inequality a true statement. Substitution can be used to determine whether a given value(s) makes an equation or inequality true.

·  Properties of real numbers and properties of equality can be used to solve equations, justify equation solutions, and express simplification. 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 neither commutative nor associative.

-  Identity property of multiplication (multiplicative identity property):

-  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.

-  Inverses are numbers that combine with other numbers and result in identity elements (e.g., 5 + (–5) = 0; · 5 = 1).

-  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.

-  Addition property of equality: If , then .

-  Subtraction property of equality: If  then .

-  Multiplication property of equality: If  then .

-  Division property of equality: If  then .

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

### ESSENTIALS

• When solving an equation, why is it necessary to perform the same operation on both sides of an equal sign?
To maintain equality, an operation performed on one side of an equation must be performed on the other side.

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

6.181  Represent and solve a one­step equation, using a variety of concrete materials such as colored chips, algeblocks, or weights on a balance scale.

·  6.132  Represent and solve one-step linear equations in one variable, using a variety of concrete materials such as colored chips, algebra tiles, or weights on a balance scale.

6.182  Solve one­step equation by demonstrating the steps algebraically.

·  6.137  Represent and solve a practical problem with a one-step linear equation in one variable.

·  6.134  Confirm solutions to one-step linear equations in one variable.

6.183  Identify and use the following algebraic terms appropriately: equation, variable, expression, term, and coefficient.

·  6.131  Identify examples of the following algebraic vocabulary: equation, variable, expression, term, and coefficient.

·  6.133  Apply properties of real numbers and properties of equality to solve a one-step equation in one variable. Coefficients are limited to integers and unit fractions. Numeric terms are limited to integers.

·  6.135  Write verbal expressions and sentences as algebraic expressions and equations.

·  6.136  Write algebraic expressions and equations as verbal expressions and sentences.

### KEY VOCABULARY

equation, linear, variable, whole number, coefficient, term, expression, equality

Updated: Oct 27, 2017