Why 5x and 5x Squared Can Never Be Added Together
Learn to spot constants, variables, terms and coefficients, classify expressions and find their degree, collect like terms by row and column methods, and evaluate an expression by substitution.
Why can 5x and 5x squared never be added together?
Because they measure different things. If is a length in metres, then is a length and is an area — and you cannot add metres to square metres any more than you can add rupees to kilograms.
That is what "unlike terms" really means. This page covers everything in the ICSE Class 7 Mathematics chapter's first part: constants and variables, terms and coefficients, classifying expressions and their degree, collecting like terms, and substitution.
That is what "unlike terms" really means. This page covers everything in the ICSE Class 7 Mathematics chapter's first part: constants and variables, terms and coefficients, classifying expressions and their degree, collecting like terms, and substitution.
What are the terms, factors and coefficients of an expression?
A constant has a fixed value, such as 7 or . A variable (or literal) can take different values and is written with a letter, such as , or .
An algebraic expression is a combination of constants and variables joined by operations. Its terms are the parts separated by or signs.
Take :
- Terms: , and — three of them.
- Factors of : , and .
- Coefficient of in the first term: 5.
- Coefficient of in the second term: .
- Constant term: , since it has no variable.
The numerical coefficient is the number part of a term, so in it is .
The perimeter of a square written as is an expression of this kind, with coefficient 4 and variable .
Two details are examined often and easy to miss. The sign belongs to the term — in the second term is , not . And when no number is written, the coefficient is 1: in , the coefficient of is 1, and in it is .
An algebraic expression is a combination of constants and variables joined by operations. Its terms are the parts separated by or signs.
Take :
- Terms: , and — three of them.
- Factors of : , and .
- Coefficient of in the first term: 5.
- Coefficient of in the second term: .
- Constant term: , since it has no variable.
The numerical coefficient is the number part of a term, so in it is .
The perimeter of a square written as is an expression of this kind, with coefficient 4 and variable .
Two details are examined often and easy to miss. The sign belongs to the term — in the second term is , not . And when no number is written, the coefficient is 1: in , the coefficient of is 1, and in it is .
How do you classify an expression and find its degree?
Expressions are named by how many terms they have:
- Monomial — one term: , , .
- Binomial — two terms: , .
- Trinomial — three terms: .
- Polynomial — an expression with one or more terms in which the variables have only whole-number powers.
The degree is the highest power of the variable present.
- has degree 3.
- has degree 1, and a constant such as has degree 0.
With more than one variable, add the powers within each term and take the largest total:
- has degree .
- has degrees and , so its degree is 5.
A volume formula such as is a monomial of degree 3, which is exactly why it measures a three-dimensional quantity.
The multi-variable rule is where marks are lost. The degree of is 3, not 2 — you add the exponents inside a term rather than picking the biggest one.
- Monomial — one term: , , .
- Binomial — two terms: , .
- Trinomial — three terms: .
- Polynomial — an expression with one or more terms in which the variables have only whole-number powers.
The degree is the highest power of the variable present.
- has degree 3.
- has degree 1, and a constant such as has degree 0.
With more than one variable, add the powers within each term and take the largest total:
- has degree .
- has degrees and , so its degree is 5.
A volume formula such as is a monomial of degree 3, which is exactly why it measures a three-dimensional quantity.
The multi-variable rule is where marks are lost. The degree of is 3, not 2 — you add the exponents inside a term rather than picking the biggest one.
How do you add and subtract algebraic expressions?
Only like terms can be combined. Like terms have exactly the same variables raised to the same powers, differing only in their coefficients.
- and are like.
- and are like.
- and are unlike — different powers.
- and are unlike — different variables.
To combine, add or subtract the coefficients and keep the variable part unchanged: .
Row method. Add and :
Column method. Write like terms one under the other and add down:
giving — the same answer, with the columns keeping the like terms aligned.
Subtraction. Change the sign of every term of the expression being subtracted, then add. Subtract from :
That sign change is the classic trap. All three signs flip, not just the first — writing would give the wrong answer, and it is worth rewriting the line before combining.
- and are like.
- and are like.
- and are unlike — different powers.
- and are unlike — different variables.
To combine, add or subtract the coefficients and keep the variable part unchanged: .
Row method. Add and :
Column method. Write like terms one under the other and add down:
giving — the same answer, with the columns keeping the like terms aligned.
Subtraction. Change the sign of every term of the expression being subtracted, then add. Subtract from :
That sign change is the classic trap. All three signs flip, not just the first — writing would give the wrong answer, and it is worth rewriting the line before combining.
How do you find the value of an expression by substitution?
Replace each variable with its given value, put brackets round it, and evaluate using BODMAS.
Worked example. Find the value of when and :
Worked example with a negative value. Find when :
Worked example with two variables. Find when and :
Substitution is how any formula becomes a number. The area of a rectangle with m and m gives , and with cm gives a perimeter of cm.
The brackets are what keep a negative value safe. In the second example , but writing would give and the whole answer would change — so substitute with brackets every time, and evaluate the powers before the multiplications.
Worked example. Find the value of when and :
Worked example with a negative value. Find when :
Worked example with two variables. Find when and :
Substitution is how any formula becomes a number. The area of a rectangle with m and m gives , and with cm gives a perimeter of cm.
The brackets are what keep a negative value safe. In the second example , but writing would give and the whole answer would change — so substitute with brackets every time, and evaluate the powers before the multiplications.
Exam tip
Exam tip: matching variables and powers exactly
Algebra marks in this chapter come from precision rather than difficulty.
Before combining anything, check that the terms have the same variables to the same powers. and cannot be added, and neither can and .
Carry the sign with the term, and remember an invisible coefficient is 1 — so has coefficients 1 and 1.
For subtraction, rewrite the expression with every sign changed on its own line before you collect anything.
For degree with several variables, add the exponents inside each term: has degree 3.
And when substituting, put each value in brackets and evaluate powers first — , not .
Before combining anything, check that the terms have the same variables to the same powers. and cannot be added, and neither can and .
Carry the sign with the term, and remember an invisible coefficient is 1 — so has coefficients 1 and 1.
For subtraction, rewrite the expression with every sign changed on its own line before you collect anything.
For degree with several variables, add the exponents inside each term: has degree 3.
And when substituting, put each value in brackets and evaluate powers first — , not .
Did you know
Why does a term with two variables have a bigger degree than it looks?
Because the degree counts how many variable factors the term contains altogether.
Write out in full and it is — three variable factors, so degree 3. The exponent 2 only tells you how many of them are .
That reading also explains what degree measures. A term of degree 1 behaves like a length, degree 2 like an area, and degree 3 like a volume — which is exactly why gives the volume of a cube while gives a perimeter.
Write out in full and it is — three variable factors, so degree 3. The exponent 2 only tells you how many of them are .
That reading also explains what degree measures. A term of degree 1 behaves like a length, degree 2 like an area, and degree 3 like a volume — which is exactly why gives the volume of a cube while gives a perimeter.
Key takeaways
Algebraic expressions: quick revision
- Constants are fixed, variables are letters; terms are separated by and , and the sign belongs to the term.
- The coefficient is the number part, and an unwritten coefficient is 1.
- Expressions are monomials, binomials, trinomials or general polynomials by number of terms.
- The degree is the highest power; with several variables, add the exponents in a term, so has degree 3.
- Only like terms — same variables, same powers — can be combined, by adding their coefficients.
- When subtracting, change the sign of every term first; when substituting, use brackets so .
You will remember all of this far better after answering five questions on it than after reading it twice.
- The coefficient is the number part, and an unwritten coefficient is 1.
- Expressions are monomials, binomials, trinomials or general polynomials by number of terms.
- The degree is the highest power; with several variables, add the exponents in a term, so has degree 3.
- Only like terms — same variables, same powers — can be combined, by adding their coefficients.
- When subtracting, change the sign of every term first; when substituting, use brackets so .
You will remember all of this far better after answering five questions on it than after reading it twice.