Hello again. Last lesson focused on reading algebra: recognizing an expression (ביטוי אלגברי), an equation (משוואה), coefficients (מקדמים), variables (משתנים), and the command word simplify (פשט).
Now you will use that language to simplify expressions. The central idea is that simplifying does not mean finding . It means rewriting an expression into an equivalent, cleaner form by:
- removing parentheses correctly with the distributive property — חוק הפילוג;
- combining like terms — איברים דומים.
For example, an expression such as can become the shorter expression , but both expressions have exactly the same value for every possible value of .
Two moves, always in this order
When an expression contains parentheses and terms that might combine, follow this reliable order:
- Distribute any number multiplying parentheses.
- Rewrite the expression without parentheses.
- Combine like terms.
- Check whether anything more can be combined.
Do not try to combine terms that are still inside parentheses with terms outside them. First remove the parentheses.
1. The distributive property — חוק הפילוג
The distributive property says that a factor outside parentheses multiplies every term inside:
For subtraction inside parentheses:
For instance:
The multiplies both and :
It is not correct to write . The was inside the parentheses, so it must also be multiplied by .
One way to understand this is repeated groups:
There are two terms and two terms, giving:
How to simplify an expression by combining like terms and the distributive property | Khan Academy
Watch “How to simplify an expression by combining like terms and the distributive property” by Khan Academy. It gives a short visual explanation of why distributing means multiplying every term, then shows the full process on an expression with subtraction.
Watch the first example to see why 2(3x+5) becomes 6x+10. Then watch the longer example, focusing especially on the moment when the subtraction before parentheses is treated as multiplication by -2.
2. Like terms — איברים דומים
Terms (איברים) are pieces separated by addition or subtraction. In
the terms are , , and .
Like terms have exactly the same variable part, including the same exponent.
| Terms | Like terms? | Why? |
|---|---|---|
| and | Yes | Both have . |
| and | Yes | Both have . |
| and | Yes | Both are constants — קבועים. |
| and | No | Different variables. |
| and | No | Different powers of . |
| and | No | The variable parts differ. |
When terms are like, combine only their coefficients:
Think of as “seven -objects.” You can add seven -objects to three -objects, but you cannot combine and , just as seven apples and three books are not the same kind of object.
A complete example
Simplify:
First, distribute the . It multiplies both terms inside the parentheses:
Now identify like terms:
- and are like terms.
- and are like terms because both are constants.
Combine them:
Therefore,
Notice that the original expression and are equivalent — שווים בערכם. They produce the same result for any .
For a quick check, choose .
Original expression:
Simplified expression:
Both give , which supports that the simplification is correct.
Signs matter: distribute carefully
A frequent mistake is losing a negative sign. Treat a negative sign as part of the number being distributed.
Positive outside, negative inside
Simplify:
Distribute :
Combine the -terms and constants:
So:
The result of is , not .
Negative outside parentheses
Now consider:
The first part is straightforward:
For the second part, read the subtraction as multiplying the entire parentheses by :
Put both parts together:
Then combine like terms:
Therefore:
The key habit is this:
When you see a minus directly before parentheses, imagine a hidden , or treat the next multiplier as negative.
For example:
It is not . The negative affects every term inside.
More than one variable or power
The method is unchanged when expressions contain several variables or powers. You still distribute first, then combine only genuinely like terms.
Simplify:
Distribute the :
Distribute the :
Write all terms together:
Group like terms:
Combine:
So:
Here, -terms can combine with -terms, and -terms can combine with -terms. But a -term and an -term stay separate.
Also be careful with powers:
There are two groups:
So:
You cannot combine with . They are different kinds of terms.
A clean written-test method
On a written test, show the important line between the original expression and the final answer. A good layout makes sign mistakes easier to catch.
For example:
Write one line for distributing:
Then one line for combining:
This is much safer than trying to do everything mentally in one jump.
Use this checklist before you finish:
- Did I multiply the outside number by every term in the parentheses?
- Did I keep the positive and negative signs?
- Did I combine only like terms?
- Did I combine all constants?
- Does my final expression have no removable parentheses and no remaining like terms?
A standard final form usually places terms with higher powers first, then lower powers, and constants last. For example:
not
Both are equivalent, but the first is the conventional, easier-to-read form.
Key takeaways
- Simplify — פשט — means rewrite an expression in an equivalent but cleaner form; it does not mean solve for a variable.
- The distributive property — חוק הפילוג — requires multiplying the outside factor by every term inside parentheses.
- Like terms — איברים דומים — have the same variable part and the same exponent. Constants are like terms with other constants.
- A minus sign before parentheses affects the entire group:
- The dependable order is: distribute, rewrite, combine like terms, and check signs.
Next, you will use these same skills to solve one-variable linear equations and verify your solution by substitution.
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