# 7.12 - Linear Equations

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

### 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

2016 VDOE Curriculum Framework - 7.12 Understanding

·  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:
2x +  = -5;  -25 = 7.2x + 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.,, 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 5” could be represented by “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 twice a number and two is fifteen.” could be represented by “2n + 2 = 15”).

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

• 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

·  7.121  Represent and solve two-step linear equations in one variable using a variety of concrete materials and pictorial representations.

·  7.122  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.

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

·  7.126  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