Combining Like Terms Notes
This Combining Like Terms Calculator shortens an algebraic expression by joining the terms that share the same variable part, and shows the working at every step. Terms are like terms only when their variable parts match exactly, so the first job is sorting them into groups. The coefficients inside each group are then added or subtracted, while the variable part is left alone. Following the same pattern every time is what makes simplifying easy to picture, check, and repeat on your own with any expression.
Step-by-step method
- Set up the problem.
- Group the like terms. Like terms have the same variable part with the same powers, so they are written next to each other, each keeping the sign in front of it.
- Combine the like terms by adding or subtracting their coefficients, keeping the variable part unchanged.
- Write the combined terms together with any unchanged terms to give the simplified expression.
See Example 1 Hide Example 1
Problem
Approach
Step-by-step method
- Set up the problem.
- Group the like terms. Like terms have the same variable part with the same powers, so they are written next to each other, each keeping the sign in front of it.
- Combine the like terms by adding or subtracting their coefficients, keeping the variable part unchanged.
- Write the combined terms together with any unchanged terms to give the simplified expression.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(3x + 2x - 5 + 7\).
Step 2
Step 2 - Group the like terms. Like terms have the same variable part with the same powers, so they are written next to each other, each keeping the sign in front of it.
In this problem: \(3x\) and \(2x\) are \(x\)-terms and \(-5\) and \(7\) are constants, and each set is already side by side, so brackets only mark the groups, giving \((3x + 2x) + (-5 + 7)\).
Step 3
Step 3 - Combine the like terms by adding or subtracting their coefficients, keeping the variable part unchanged.
In this problem: There are \(2\) groups of like terms to combine: the \(x\)-terms and the constants.
Step 3a
Step 3a - Combine the \(x\)-terms.
In this problem: The coefficients of the \(x\)-terms are \(3\) and \(2\), and \(3 + 2 = 5\), so \(3x + 2x = 5x\).
Step 3b
Step 3b - Combine the constants.
In this problem: The constants are \(-5\) and \(7\), and \(-5 + 7 = 2\).
Step 4
Step 4 - Write the combined terms together with any unchanged terms to give the simplified expression.
In this problem: Putting each result back in place gives \(5x + 2\).
Final Answer
See Example 2 Hide Example 2
Problem
Approach
Step-by-step method
- Set up the problem.
- Group the like terms. Like terms have the same variable part with the same powers, so they are written next to each other, each keeping the sign in front of it.
- Combine the like terms by adding or subtracting their coefficients, keeping the variable part unchanged.
- Write the combined terms together with any unchanged terms to give the simplified expression.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(4x^{2} + 3x - x^{2} + 5 - 2x\).
Step 2
Step 2 - Group the like terms. Like terms have the same variable part with the same powers, so they are written next to each other, each keeping the sign in front of it.
In this problem: \(4x^{2}\) and \(-x^{2}\) are \(x^{2}\)-terms and \(3x\) and \(-2x\) are \(x\)-terms, so each set is written together, giving \((4x^{2} - x^{2}) + (3x - 2x) + 5\).
Step 3
Step 3 - Combine the like terms by adding or subtracting their coefficients, keeping the variable part unchanged.
In this problem: There are \(2\) groups of like terms to combine: the \(x^{2}\)-terms and the \(x\)-terms.
Step 3a
Step 3a - Combine the \(x^{2}\)-terms.
In this problem: The coefficients of the \(x^{2}\)-terms are \(4\) and \(-1\), and \(4 - 1 = 3\), so \(4x^{2} - x^{2} = 3x^{2}\).
Step 3b
Step 3b - Combine the \(x\)-terms.
In this problem: The coefficients of the \(x\)-terms are \(3\) and \(-2\), and \(3 - 2 = 1\), so \(3x - 2x = x\).
Step 4
Step 4 - Write the combined terms together with any unchanged terms to give the simplified expression.
In this problem: Putting each result back in place, with \(5\) unchanged, gives \(3x^{2} + x + 5\).
Final Answer
See Example 3 Hide Example 3
Problem
Approach
Step-by-step method
- Set up the problem.
- Group the like terms. Like terms have the same variable part with the same powers, so they are written next to each other, each keeping the sign in front of it.
- Combine the like terms by adding or subtracting their coefficients, keeping the variable part unchanged.
- Write the combined terms together with any unchanged terms to give the simplified expression.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(6y - 2y + 3 - 8\).
Step 2
Step 2 - Group the like terms. Like terms have the same variable part with the same powers, so they are written next to each other, each keeping the sign in front of it.
In this problem: \(6y\) and \(-2y\) are \(y\)-terms and \(3\) and \(-8\) are constants, and each set is already side by side, so brackets only mark the groups, giving \((6y - 2y) + (3 - 8)\).
Step 3
Step 3 - Combine the like terms by adding or subtracting their coefficients, keeping the variable part unchanged.
In this problem: There are \(2\) groups of like terms to combine: the \(y\)-terms and the constants.
Step 3a
Step 3a - Combine the \(y\)-terms.
In this problem: The coefficients of the \(y\)-terms are \(6\) and \(-2\), and \(6 - 2 = 4\), so \(6y - 2y = 4y\).
Step 3b
Step 3b - Combine the constants.
In this problem: The constants are \(3\) and \(-8\), and \(3 - 8 = -5\).
Step 4
Step 4 - Write the combined terms together with any unchanged terms to give the simplified expression.
In this problem: Putting each result back in place gives \(4y - 5\).
Final Answer
See Example 4 Hide Example 4
Problem
Approach
Step-by-step method
- Set up the problem.
- Group the like terms. Like terms have the same variable part with the same powers, so they are written next to each other, each keeping the sign in front of it.
- Combine the like terms by adding or subtracting their coefficients, keeping the variable part unchanged.
- Write the combined terms together with any unchanged terms to give the simplified expression.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(5a + 4b - 2a - b + 6\).
Step 2
Step 2 - Group the like terms. Like terms have the same variable part with the same powers, so they are written next to each other, each keeping the sign in front of it.
In this problem: \(5a\) and \(-2a\) are \(a\)-terms and \(4b\) and \(-b\) are \(b\)-terms, so each set is written together, giving \((5a - 2a) + (4b - b) + 6\).
Step 3
Step 3 - Combine the like terms by adding or subtracting their coefficients, keeping the variable part unchanged.
In this problem: There are \(2\) groups of like terms to combine: the \(a\)-terms and the \(b\)-terms.
Step 3a
Step 3a - Combine the \(a\)-terms.
In this problem: The coefficients of the \(a\)-terms are \(5\) and \(-2\), and \(5 - 2 = 3\), so \(5a - 2a = 3a\).
Step 3b
Step 3b - Combine the \(b\)-terms.
In this problem: The coefficients of the \(b\)-terms are \(4\) and \(-1\), and \(4 - 1 = 3\), so \(4b - b = 3b\).
Step 4
Step 4 - Write the combined terms together with any unchanged terms to give the simplified expression.
In this problem: Putting each result back in place, with \(6\) unchanged, gives \(3a + 3b + 6\).
Final Answer