The ability to simplify and solve mathematical equations by combining like terms is a fundamental skill in algebra. Whether you're a student wrestling with homework or a professional dealing with mathematical modeling, understanding how to combine terms effectively can make complex problems much more manageable. Let's walk through 5 easy steps to master this technique.
Step 1: Identify Like Terms
The first step in combining terms is to identify which terms in your equation are 'like'. Like terms are expressions that have the same variable(s) raised to the same power. For example:
- 3x and 5x are like terms because they both have x to the power of 1.
- 7y and y are like terms.
- -4a2 and 2a2 are also like terms since they have the same variable and exponent.
🔎 Note: Be cautious of coefficients and signs; they must be considered when identifying like terms.
Step 2: Rearrange the Equation
After identifying like terms, rearrange the equation in a way that groups like terms together. This step might not always be necessary if your equation is already well-organized, but it can save time and reduce confusion:
- Move all the x terms to one side of the equation.
- Group y terms together.
- Place constants (terms without variables) together.
📝 Note: When moving terms, remember to change their sign appropriately.
Step 3: Combine the Like Terms
Once like terms are grouped, simply add or subtract their coefficients:
- If you have 3x + 5x, combine them to get 8x.
- If you see -4a2 + 2a2, the result is -2a2.
Consider this example:
3x + 2y + 7 - 5x - 4y - 1
| Initial Equation | 3x + 2y + 7 - 5x - 4y - 1 |
|---|---|
| Grouped Like Terms | (3x - 5x) + (2y - 4y) + (7 - 1) |
| After Combining | -2x - 2y + 6 |
✅ Note: Double-check your work to ensure no like terms are missed.
Step 4: Solve the Equation
With the equation now simplified, solve for the variable:
- If your simplified equation is -2x - 2y + 6 = 0, you can solve for x if y is given or vise versa.
- If your equation has constants only, the solution is straightforward.
🔧 Note: For systems of equations, you might need to use substitution or elimination methods after combining like terms.
Step 5: Verify Your Solution
The final step is to verify your solution:
- Plug the found variable back into the original equation.
- Check if both sides of the equation balance.
For instance, if you found x = 2, substitute it back:
3(2) + 2y + 7 - 5(2) - 4y - 1
6 + 2y + 7 - 10 - 4y - 1 = 2y - 4y + 12 - 11 = -2y + 1
If the left side equals the right side, your solution is correct.
In closing, mastering the art of combining like terms not only simplifies the process of solving algebraic equations but also enhances your understanding of the underlying structure of equations. Through practice, these steps will become second nature, allowing you to navigate through algebraic problems with ease.
What are like terms?
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Like terms are terms that have the same variable(s) raised to the same power.
How can I simplify if there are multiple variables?
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Group and combine like terms for each variable separately before solving for individual variables.
Why is it important to rearrange equations?
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Rearranging equations can make combining like terms more straightforward by grouping terms together logically.
