Lesson Objectives

  • Learn how to Identify Provisional Equations
  • Acquire how to Identify an Identity Equation
  • Learn how to Identify a Contradiction Equation

How to Place the Type of Equation as: Provisional, Identity, or Contradiction

As nosotros learned in Algebra 1, we can categorize our equations as: provisional, identity, or contradiction.

Conditional Equations

Most equations we work with are conditional equations. A conditional equation is true only under certain atmospheric condition. Let's have a await at an case.
Example 1: Make up one's mind if the equation is conditional, an identity, or a contradiction.
-8x - 38 = -5(ane - 5x)
Allow's brainstorm by removing parentheses on the right side:
-8x - 38 = -5 + 25x
Now nosotros tin movement all the variable terms to the left side and all the constants to the right side:
-8x - 25x = -v + 38
-33x = 33
Divide each side of the equation past -33, this will isolate x:
(-33/-33)x = (33/-33)
x = -1
Since we have only one solution, we tin say this equation is a provisional equation. It is true when (-1) replaces 10, simply false for whatsoever other number.

Contradiction Equation

From fourth dimension to time, we will run across an equation with no solution. This equation type is known every bit a contradiction. Let's take a look at an instance.
Example 2: Make up one's mind if the equation is conditional, an identity, or a contradiction.
-4x - 13 = -(8 + 4x)
Allow's begin by removing parentheses on the right side:
-4x - xiii = -8 - 4x
At present we can motion all the variable terms to the left side and all the constants to the right side:
-4x + 4x = -8 + 13
0 = five (false)
When we stop upward with a false argument, we can stop and say our equation has "no solution". Since we studied sets earlier in our course, nosotros can also use the empty set symbol to bear witness our solution fix is empty, meaning it contains no elements:
∅

Identity Equation

Lastly, we will run into equations that have an infinite number of solutions. Allow's take a expect at an example.
Example 3: Determine if the equation is conditional, an identity, or a contradiction.
2x + v(ten - 8) = -40 + 7x
Let's begin past removing parentheses on the left side:
2x + 5x - xl = -twoscore + 7x
Now we can simplify the left side:
7x - 40 = -40 + 7x
Earlier we become any farther, we should discover that the two sides have exactly the same terms (7x and -40). This ways whatever value is plugged in for x volition always yield a true statement.
If nosotros keep with our normal procedure, we would next motion all the variable terms to the left and all the constants to the right:
7x - 7x = -xl + 40
0 = 0 (true)
When we end upward with a truthful statement and no variable, nosotros can stop and say our equation has "an infinite number of solutions".

Skills Check:

Case #1

Identify the type of equation. $$31 - 5x=-five(x - six)$$

Delight cull the all-time answer.

Example #two

Identify the equation, solve if possible. $$25 + ten=4(one - 5x)$$

Please choose the best reply.

Case #3

Place the equation, solve if possible. $$-30 + 5x=5(x - 6)$$

Please cull the best respond.

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