Solving Equations with Variables on Both Sides of the Equation
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GOAL: I can solve problems that have variables on both sides of the equation.
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| Text reference: Chapter 2, section 4, pages: 102 ─ 108 |
Illinois learning standard: 8.C.3 |
NCTM standard: XX─XX |
| Lesson Vocabulary |
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Identity Equation
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An identity equation is a equation that when solved, has the exact same value on both sides of the equation. For example, if we simplify the following:
2x + 5 = 5x + 5 ─ 3x
When we combine the 5x and (─3x) on the right side of the equation, we get:
2x + 5 = 2x + 5
The exact same binomial is on the left side of the equation as on the right side of the equation. This is an identity equation.
ANY value that is substituted in place of the variable "x" will make this a true equation. That means that every value in the universe of numbers is an answer to this problem.
And since there is NOT just one unique answer to this equation, it is called an identity equation.
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| Internet Video Tutorials | | | | Topic | Problem | Time (min: sec) | Solving Equations with Variables on Both Sides of the Equation
| 20 ─ 7y = 6y ─ 6
| 4:33 | | Solving Equations with Variables on Both Sides of the Equation | x ─ 8 = (x/3) + (1/6)
| 5: 46
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Strategies and Content Practice |
Notes |
Begin
(prepare) |
Check for prior knowledge.
Discuss and review solving multi-step problems.
Discuss and review using the distributive property.
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| Engage |
Have students read pages 102 ─ 106. Read aloud same with students.
Demonstrate the identity equation example given in the definition section.
Equations with variables on both sides an extension of solving multi-step equations. The techniques used in solving multi-step equations generally apply to only one side of the equation. In this lesson, the techniques for solving multi-step equations are applied to both sides of the equation. In short, solving problems with variables on both sides of the equation simply requires an additional simplification step or two.
Three outcomes or "cases" will occur when solving problems with variables on both sides of the equal sign. These cases are explained below.
Case #1
In case #1, the equation can be solved for a numeric value. This is normally what you might expect to happen.
Case #2 and Case #3
In case #2 and case #3, the variables on both sides of the equal sign turn out to be identical. When variables on both sides of the equal sign are identical, they cancel each other out and disappear from the equation. What is left over after the variables are gone?
Case #2
In case #2, the variables on both sides of the equal sign were identical and they cancelled each other out.
7x + 8 = 7x + (─4)
(7x cancels from both sides leaving...)
8 ≠ (─4) or "NO SOLUTION"
There are now two numbers on both sides of the equal sign. If the leftover numbers on the left and right sides of the equation are not identical (different values), then we call this case a "NO SOLUTION" case.
Case #3
In case #3, the variables on both sides of the equal sign were identical and they cancelled each other out.
─2x + 5 = ─2x + 5
(─2x cancels from both sides leaving...)
5 = 5 or "IDENTITY"
There are now two numbers on both sides of the equal sign. If the
leftover numbers on the left and right sides of the equation are not
identical (different values), then we call this case a "IDENTITY" case.
Show demonstration videos for Lesson 2─4 (above). |
Try this online Algebra Tiles App to practice solving problems with variables on both sides of the equal sign. |
Assess
(formative) |
After example problems and videos, have students try to solve the Sample Problems below. After students have had time to complete them, click on the Solutions link and go through the step-by-step process for solving each problem. Repeat if necessary with additional example problems in the book and re-assess. |
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| Apply |
Click HERE to download the Practice Assignment.
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Depending on class conditions, you may allow students to work with each other.
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Assess
(formative) |
Click HERE to download the Answers to the Practice Assignment. | Depending on level of understanding, you may accept this pre-assessment grade and allow students who have mastered these concepts to skip the formal assessment exercise. | | Apply | Click HERE to download the Graded Assignment.
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| Sample Problems |
Sample Solutions |
5x + 2 = 2x + 14
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Click Here for step-by-step solution. |
3x + 4 = 5x ─ 10
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Click Here for step-by-step solution. |
| 5(x ─ 4) = 7(2x + 1) |
Click Here for step-by-step solution. |
| 2x + 3 = ½(4x + 3) |
Click Here for step-by-step solution. |
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