# 8.17 - Equations

8.17  The student will solve multistep linear equations in one variable with the variable on one or both sides of the equation, including practical problems that require the solution of a multistep linear equation in one variable.

### BIG IDEAS

• I can find how far and how fast a bus travels, the number and combinations of fruits that can be purchased, and how long it takes to drain a swimming pool.  I can avoid receiving a speeding ticket when I can drive and won’t go over the allowed number of text messages per month on my cell phone.
• I can translate real-life situations into equations to find unknown values.
• 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

2016 VDOE Curriculum Framework - 8.17 Understanding

·  A multistep equation may include, but not be limited to equations  such as the following:  = ; ;

·  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. For example, the verbal expression “a number multiplied by five” 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., 2x – 3).

·  A verbal sentence is a complete word statement (e.g., “The sum of two consecutive integers is thirty-five.” could be represented by “n+ (n + 1) = 35”).

·  An algebraic equation is a mathematical statement that says that two expressions are equal
(e.g., 2x + 3 = -4x +1).

·  In an equation, the “equal sign (=)” indicates that the value of the expression on the left is equivalent to the value of the expression on the right.

·  Like terms are terms that have the same variables and exponents. The coefficients do not need to match (e.g., 12x and −5x; 45 and −5 ; 9y, −51y and y.)

·  Like terms may be added or subtracted using the distributive and other properties. For example,

-  4.6y – 5y = (−4.6 – 5)y = −9.6y

-  w + w – 2w = (1 + 1)w – 2w = 2w – 2w = (2 – 2)w = 0w = 0

·  Real-world problems can be interpreted, represented, and solved using linear equations in one variable.

·  Properties of real numbers and properties of equality can be used to solve equations, justify 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: .

-  Associative property of addition: .

-  Associative property of multiplication: .

-  Subtraction and division are neither commutative nor associative.

.

-  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 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 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

• How does the solution to an equation differ from the solution to an inequality?
While a linear equation has only one replacement value for the variable that makes the equation true, an inequality can have more than one.

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

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

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

·  8.171  Represent and solve multistep linear equations in one variable with the variable on one or both sides of the equation (up to four steps) using a variety of concrete materials and pictorial representations. Apply properties of real numbers and properties of equality to solve multistep linear equations in one variable (up to four steps). Coefficients and numeric terms will be rational.  Equations may contain expressions that need to be expanded (using the distributive property) or require collecting like terms to solve.

·  8.174  Solve practical problems that require the solution of a multistep linear equation.

·  8.175  Confirm algebraic solutions to linear equations in one variable.

### KEY VOCABULARY

linear equation, pictorial representations, two step inequalities, algebraic sentences, commutative property of addition and multiplication, associative property of addition and multiplication, distributive property, identity property of addition and multiplication, zero property of multiplication, additive inverse, multiplicative inverse

Updated: Jul 30, 2019