~ 6.20 ~

Combining Like Terms (Part 1)

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Learning Targets

I can figure out whether two expressions are equivalent to each other.
When possible, I can write an equivalent expression that has fewer terms.

Notes

There are many ways to write equivalent expressions that may look very different from each other. We have several tools to find out if two expressions are equivalent.

Two expressions are definitely not equivalent if they have different values when we substitute the same number for the variable. For example, 2(-3 + x) + 8 and 2x + 5 are not equivalent because when x is 1, the first expression equals 4 and the second expression equals 7.
If two expressions are equal for many different values we substitute for the variable, then the expressions may be equivalent, but we don't know for sure. It is impossible to compare the two expressions for all values. To know for sure, we use properties of operations. For example, 2(-3 + x) + 8 is equivalent to 2x + 2 because:

2(-3 + x ) + 8

-6 + 2x + 8 by the distributive property

2x + -6 + 8 by the distributive property

2x + (-6 + 8) by the associative property

2x + 2

Combining like terms with negative coefficientsSimplify algebraic expressions by combining like terms. Coefficients on some terms are negative.

Activities

20.1 Why is it True?

Explain why each statement is true.

5 + 2 + 3 = 5 + (2 + 3)
9a is equivalent to 11a - 2a
7a + 4 - 2a is equivalent to 7a + -2a + 4
8a - (8a - 8) is equivalent to 8

20.2 A’s and B’s

Diego and Jada are both trying to write an expression with fewer terms that is equivalent to 7a + 5b - 3a + 4b.

Jada thinks 10a + 1b is equivalent to the original expression.
Diego thinks 4a + 9b is equivalent to the original expression.

Add to Your Notes

We can tell whether the expressions are equivalent by substituting some different values for a and b and evaluating the expressions.

Let’s try a = 4 and b = 3.

Now try a different set of numbers for a and b.

Experimenting the numbers can tell us that two expressions are not equivalent, but can’t prove that two expression are equivalent.

Instead, we need to think about the expressions using the properties that we know.

We can show expressions are equivalent by writing out all the variables. Explain why the expression on each row (after the first row) is equivalent to the expression on the row before it.

2. Here is another way we can rewrite the expressions. Explain why the expression on each row (after the first row) is equivalent to the expression on the row before it.

Add to Your Notes

Which method do you prefer for deciding whether expressions are equivalent? Substituting values or using the properties of operations?

While checking values can give us useful information, there is usually no way to check all possible values. That’s why we need to have some algebraic methods to rely on!

20.3 Making Sides Equal

Replace each ? with an expression that will make the left side of the equation equivalent to the right side.

Set A

6x + ? = 10x
6x + ? = 2x
6x + ? = -10x
6x + ? = 0
6x + ? = 10

Set B

6x - ? = 2x
6x - ? = 10x
6x - ? = x
6x - ? = 6
6x - ? = 4x - 10

Summary

Assignment

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