Equation Basics
Which of the following is an equation? Why?
A) 6x - 4
B) 6x - 4 = 20
Answer: B
Explanation: An equation must include an equal sign.
What is the first step to solve:
x + 9 = 21
Answer: Subtract 9 from both sides
Explanation: Undo addition first.
Which equation is an identity? Why?
A) 2x + 7 = 2x + 7
B) 3x + 1 = 3x + 5
Answer: A
Explanation: True for all values of x. Identity = Identical
Solve for b:
a+b=18
Answer: b=18−a
OR b = -a+18
Explanation: Undo addition with subtraction to get b by itself on one side of the equal sign/equation.
Translate:
“Three more than a number, x, is 12.”
Answer: x + 3 = 12
The formula C = 2d/a is solved for which variable?
Answer: C
Explanation: The variable isolated on one side of the equal sign is the solution variable.
What could be the first step in solving:
4(3x−2)=12x−8
Answer: Distribute the 4
Explanation: Eliminate parentheses first.
Which equation is a contradiction?
A) 4x − 6 = 4x − 6
B) 5x + 2 = 5x + 9
Answer: B
Explanation: Leads to a false statement.
Solve for x:
5y+x=30
Answer: x=30−5y
OR x = -5y + 30
Translate:
“Five less than twice a number, n, is 9.”
Answer: 2n - 5 = 9
Which equation is already solved for y?
A) 4y - x = 10
B) x = 3y + 2
C) y = 5x - 7
D) 2x + y = 9
Explanation: y is isolated on one side.
Which is NOT a valid next step when solving:
6x+4=2x+20
Answer: Dividing both sides by 2 or 6
Explanation: You must combine like terms to one side (move the variables and constant terms to either side with addition/subtraction)
Solve and classify:
6(x+4)=6x+24
Answer: Identity
Explanation: Both sides simplify to the same expression.
Solve for r:
A = pir^2
Answer =
r=sqrt(A/pi)
Write let statements:
“One number is 4 more than another.”
Answer:
Let x = first number
Let x + 4 = second number
Which of the following could be an equation but is written incorrectly? Why?
A) 4x + 9
B) y + 7 =
C) 3x = 18
D) 2(5x-1)
Answer: B
Explanation: An equation must have two complete expressions, one on each side of the equal sign.
Choice B includes an equal sign but is incomplete, so it could be an equation but is written incorrectly.
What operation should be undone first when solving:
5x−11=29
Answer: Subtraction
Explanation: Undo addition/subtraction before multiplication/division.
Solve and classify:
3(x+5)=3x+12
Answer: Contradiction
Explanation: Simplifies to 15 = 12 (false).
Why do we isolate the variable?
Answer: To find its value (what solutions will make the math statement/equation true)
Explanation: Solving means getting the variable alone.
The total cost is $640. Parts cost $280. Labor is $40 per hour.
Let h = hours. Write the equation.
Answer: 280 + 40h = 640
Determine which statement best describes the equation:
4(x−3)+2=4x−10
A) The equation has exactly one solution
B) The equation is an identity with infinitely many solutions
C) The equation is a contradiction with no solution
D) The equation cannot be classified without graphing
Answer: B
Explanation:
Distribute and simplify mentally:
4x−12+2=4x−10⇒
4x−10=4x−10
Both sides are identical, so the equation is an identity, meaning all real numbers are solutions.
When solving the equation
3(2x−5)+4=2(3x+1)
which choice identifies the best first step and explains why it is the best strategy?
A) Subtract 2(3x+1)2(3x + 1)2(3x+1) from both sides to move variables together
B) Distribute on both sides to eliminate parentheses
C) Divide both sides by 3 to simplify the coefficients
D) Subtract 4 from both sides to isolate the variable term
Answer: B
Explanation:
The equation contains parentheses on both sides. Distributing first removes the parentheses and reveals like terms, making it easier to combine terms and solve. Undoing operations like subtraction or division too early would make the equation more complicated rather than simpler.
What does an identity and contradiction mean in words?
Answer: Identity = All real numbers are solutions; Contradiction = No real solutions.
Explanation: Identity= The equation is always true. Contradiction = the equation is always false.
Solve for b: 2a + 3b = c
Answer:
b = (c-2a)/3
Eight more than twice a number is 142.
What is the number?
Solve and answer in a complete sentence.
(hint: MUST have a let statement)
Answer:
Let x = the number
Translation (equation): 2x + 8 = 142
Solve: Subtract 8 from both sides of the equation.
2x = 134
Divide both sides of the equation by 2.
x = 67
Complete sentence: The number is 67.
Explanation:
The phrase “twice a number” translates to 2x, “eight more than” means add 8, and “is” means equals. Solving the equation gives the value of the number.