Simplifying Algebraic Expressions Notes
This Simplifying Algebraic Expressions Calculator shortens an expression by working through it in the order the operations demand, and shows the working at every step. Brackets come first, so each one is opened by multiplying the term outside it through the terms inside. Once no brackets are left the expression is a plain sum, and the terms sharing a variable part are joined into one. 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.
- Open every bracket first, because brackets come before anything else in the order of operations. Do a distributive property on each one.
- With the brackets gone the expression is a plain sum, so do a combining like terms to join the terms that match.
See Example 1 Hide Example 1
Problem
Approach
Step-by-step method
- Set up the problem.
- Open every bracket first, because brackets come before anything else in the order of operations. Do a distributive property on each one.
- With the brackets gone the expression is a plain sum, so do a combining like terms to join the terms that match.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(4x + 3\left(2x - 5\right) - 7\), which contains brackets.
Step 2
Step 2 - Open every bracket first, because brackets come before anything else in the order of operations. Do a distributive property on each one.
In this problem: Opening the brackets in \(4x + 3\left(2x - 5\right) - 7\) turns it into \(4x + 6x - 15 - 7\), worked through below.
Problem
Approach
Step-by-step method
- Set up the problem.
- Write the formula.
- Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
- Do a combining like terms on what is left.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(4x + 3\left(2x - 5\right) - 7\), with \(1\) bracket to open.
Step 2
Step 2 - Write the formula.
In this problem: Every bracket follows \(a\left(b + c\right) = ab + ac\), with the sign in front of a term travelling with it. Here \(a = 3\), \(b = 2x\) and \(c = -5\).
Step 3
Step 3 - Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
In this problem: The \(3\) in front of \(\left(2x - 5\right)\) multiplies every term inside it: \(3 \cdot 2x = 6x\) and \(3 \cdot \left(-5\right) = -15\).
Final Answer
Step 3
Step 3 - With the brackets gone the expression is a plain sum, so do a combining like terms to join the terms that match.
In this problem: In \(4x + 6x - 15 - 7\) the terms that share a variable part are joined, which gives \(10x - 22\), worked through below.
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 \(4x + 6x - 15 - 7\).
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: \(4x\) and \(6x\) are \(x\)-terms and \(-15\) and \(-7\) are constants, and each set is already side by side, so brackets only mark the groups, giving \((4x + 6x) + (-15 - 7)\).
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 \(4\) and \(6\), and \(4 + 6 = 10\), so \(4x + 6x = 10x\).
Step 3b
Step 3b - Combine the constants.
In this problem: The constants are \(-15\) and \(-7\), and \(-15 - 7 = -22\).
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 \(10x - 22\).
Final Answer
Final Answer
See Example 2 Hide Example 2
Problem
Approach
Step-by-step method
- Set up the problem.
- Open every bracket first, because brackets come before anything else in the order of operations. Do a distributive property on each one.
- With the brackets gone the expression is a plain sum, so do a combining like terms to join the terms that match.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(3\left(x + 4\right) - 2\left(y - 5\right)\), which contains brackets.
Step 2
Step 2 - Open every bracket first, because brackets come before anything else in the order of operations. Do a distributive property on each one.
In this problem: Opening the brackets in \(3\left(x + 4\right) - 2\left(y - 5\right)\) turns it into \(3x + 12 - 2y + 10\), worked through below.
Problem
Approach
Step-by-step method
- Set up the problem.
- Write the formula.
- Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
- Do a combining like terms on what is left.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(3\left(x + 4\right) - 2\left(y - 5\right)\), with \(2\) brackets to open.
Step 2
Step 2 - Write the formula.
In this problem: Every bracket follows \(a\left(b + c\right) = ab + ac\), with the sign in front of a term travelling with it. In bracket \(1\), \(a = 3\), \(b = x\) and \(c = 4\). In bracket \(2\), \(a = -2\), \(b = y\) and \(c = -5\).
Step 3
Step 3 - Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
In this problem: The brackets here are \(3\left(x + 4\right)\) and \(-2\left(y - 5\right)\). They do not feed into each other, so the order does not change the answer and they are opened left to right.
Step 3a
Step 3a - Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
In this problem: The \(3\) in front of \(\left(x + 4\right)\) multiplies every term inside it: \(3 \cdot x = 3x\) and \(3 \cdot 4 = 12\).
Step 3b
Step 3b - Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
In this problem: The \(-2\) in front of \(\left(y - 5\right)\) multiplies every term inside it: \(\left(-2\right) \cdot y = -2y\) and \(\left(-2\right) \cdot \left(-5\right) = 10\).
Final Answer
Step 3
Step 3 - With the brackets gone the expression is a plain sum, so do a combining like terms to join the terms that match.
In this problem: In \(3x + 12 - 2y + 10\) the terms that share a variable part are joined, which gives \(3x + 22 - 2y\), worked through below.
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 \(3x + 12 - 2y + 10\).
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: \(12\) and \(10\) are constants, so each set is written together, giving \(3x + (12 + 10) - 2y\).
Step 3
Step 3 - Combine the like terms by adding or subtracting their coefficients, keeping the variable part unchanged.
In this problem: The constants are \(12\) and \(10\), and \(12 + 10 = 22\).
Step 4
Step 4 - Write the combined terms together with any unchanged terms to give the simplified expression.
In this problem: Putting the result back in place, with \(3x\) and \(-2y\) unchanged, gives \(3x + 22 - 2y\).
Final Answer
Final Answer
See Example 3 Hide Example 3
Problem
Approach
Step-by-step method
- Set up the problem.
- Open every bracket first, because brackets come before anything else in the order of operations. Do a distributive property on each one.
- With the brackets gone the expression is a plain sum, so do a combining like terms to join the terms that match.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(2\left(3x + 1\right) + 4x - 5\), which contains brackets.
Step 2
Step 2 - Open every bracket first, because brackets come before anything else in the order of operations. Do a distributive property on each one.
In this problem: Opening the brackets in \(2\left(3x + 1\right) + 4x - 5\) turns it into \(6x + 2 + 4x - 5\), worked through below.
Problem
Approach
Step-by-step method
- Set up the problem.
- Write the formula.
- Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
- Do a combining like terms on what is left.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(2\left(3x + 1\right) + 4x - 5\), with \(1\) bracket to open.
Step 2
Step 2 - Write the formula.
In this problem: Every bracket follows \(a\left(b + c\right) = ab + ac\), with the sign in front of a term travelling with it. Here \(a = 2\), \(b = 3x\) and \(c = 1\).
Step 3
Step 3 - Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
In this problem: The \(2\) in front of \(\left(3x + 1\right)\) multiplies every term inside it: \(2 \cdot 3x = 6x\) and \(2 \cdot 1 = 2\).
Final Answer
Step 3
Step 3 - With the brackets gone the expression is a plain sum, so do a combining like terms to join the terms that match.
In this problem: In \(6x + 2 + 4x - 5\) the terms that share a variable part are joined, which gives \(10x - 3\), worked through below.
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 \(6x + 2 + 4x - 5\).
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: \(6x\) and \(4x\) are \(x\)-terms and \(2\) and \(-5\) are constants, so each set is written together, giving \((6x + 4x) + (2 - 5)\).
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 \(x\)-terms and the constants.
Step 3a
Step 3a - Combine the \(x\)-terms.
In this problem: The coefficients of the \(x\)-terms are \(6\) and \(4\), and \(6 + 4 = 10\), so \(6x + 4x = 10x\).
Step 3b
Step 3b - Combine the constants.
In this problem: The constants are \(2\) and \(-5\), and \(2 - 5 = -3\).
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 \(10x - 3\).
Final Answer
Final Answer
See Example 4 Hide Example 4
Problem
Approach
Step-by-step method
- Set up the problem.
- Open every bracket first, because brackets come before anything else in the order of operations. Do a distributive property on each one.
- With the brackets gone the expression is a plain sum, so do a combining like terms to join the terms that match.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(5\left(a - 2\right) - 3\left(a + 1\right)\), which contains brackets.
Step 2
Step 2 - Open every bracket first, because brackets come before anything else in the order of operations. Do a distributive property on each one.
In this problem: Opening the brackets in \(5\left(a - 2\right) - 3\left(a + 1\right)\) turns it into \(5a - 10 - 3a - 3\), worked through below.
Problem
Approach
Step-by-step method
- Set up the problem.
- Write the formula.
- Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
- Do a combining like terms on what is left.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(5\left(a - 2\right) - 3\left(a + 1\right)\), with \(2\) brackets to open.
Step 2
Step 2 - Write the formula.
In this problem: Every bracket follows \(a\left(b + c\right) = ab + ac\), with the sign in front of a term travelling with it. In bracket \(1\), \(a = 5\), \(b = a\) and \(c = -2\). In bracket \(2\), \(a = -3\), \(b = a\) and \(c = 1\).
Step 3
Step 3 - Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
In this problem: The brackets here are \(5\left(a - 2\right)\) and \(-3\left(a + 1\right)\). They do not feed into each other, so the order does not change the answer and they are opened left to right.
Step 3a
Step 3a - Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
In this problem: The \(5\) in front of \(\left(a - 2\right)\) multiplies every term inside it: \(5 \cdot a = 5a\) and \(5 \cdot \left(-2\right) = -10\).
Step 3b
Step 3b - Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
In this problem: The \(-3\) in front of \(\left(a + 1\right)\) multiplies every term inside it: \(\left(-3\right) \cdot a = -3a\) and \(\left(-3\right) \cdot 1 = -3\).
Final Answer
Step 3
Step 3 - With the brackets gone the expression is a plain sum, so do a combining like terms to join the terms that match.
In this problem: In \(5a - 10 - 3a - 3\) the terms that share a variable part are joined, which gives \(2a - 13\), worked through below.
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 \(5a - 10 - 3a - 3\).
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 \(-3a\) are \(a\)-terms and \(-10\) and \(-3\) are constants, so each set is written together, giving \((5a - 3a) + (-10 - 3)\).
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 constants.
Step 3a
Step 3a - Combine the \(a\)-terms.
In this problem: The coefficients of the \(a\)-terms are \(5\) and \(-3\), and \(5 - 3 = 2\), so \(5a - 3a = 2a\).
Step 3b
Step 3b - Combine the constants.
In this problem: The constants are \(-10\) and \(-3\), and \(-10 - 3 = -13\).
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 \(2a - 13\).
Final Answer
Final Answer