Algebraic expressions

In \(3x+7\), \(x\) is the variable because its value can change. The number 3 is the coefficient of \(x\), while 7 is the constant term; both are fixed numbers. Exam tip: A letter that can take different values is usually the variable.

A constant term is a term that contains no variable. In 5y - 9, the term 5y contains the variable y, whereas -9 has no variable. Therefore, -9 is the constant term. Exam tip: Identify the term with no variable to find the constant term.

A coefficient is the number that multiplies a variable. It tells us how many times the variable is being taken. In a term such as 8a, the letter a is the variable and 8 is its numerical multiplier. Therefore, the coefficient of a is 8. The coefficient is not the letter a itself, because a names the variable. It is also not 1, although a variable without a written number, such as a, has an understood coefficient of 1.

To identify the coefficient, separate the numerical part from the variable part. In 8a, the numerical part appears before a and is multiplied by it. Hence option A is correct. No addition, subtraction, or further calculation is needed; simply reading the product carefully gives the answer.

Parts of an expression separated by plus (+) or minus (−) signs are called terms. Here, the terms are p, q, and 4, so there are 3 terms. Although 4 is a constant term, it is still counted as a separate term. Exam tip: To count terms, split the expression at + and − signs.

\(7m\) has only one term, so it is a monomial. \(m+7\) and \(m-n\) are binomials, while \(m+n+7\) has three terms and is a trinomial. Exam tip: Count terms by separating an expression at addition or subtraction signs.

A binomial is an algebraic expression with two unlike terms, usually separated by a + or − sign. In x+6, the terms are x and 6, so it is a binomial. 4x and 9 are monomials, while x+y+z is a trinomial. Exam tip: Count the parts separated by + or − signs to identify the number of terms.

A trinomial is an algebraic expression with three terms separated by addition or subtraction signs. In \(a+b+c\), the terms are \(a\), \(b\), and \(c\), so it is a trinomial. \(a-5\) has only two terms, so it is a binomial. Exam tip: Count the parts separated by + or − signs to find the number of terms.

\(4x\) and \(3x\) are like terms because both have the variable \(x\) raised to power 1. Therefore, add their coefficients: \(4+3=7\). Hence, \(4x+3x=7x\). \(7x^2\) is incorrect because addition does not change the exponent of the variable. Exam tip: Add or subtract only terms with the same variables and the same powers.

Here, 9y and 2y are like terms because both contain y to the power 1. Therefore, subtract their coefficients: \(9y-2y=(9-2)y=7y\). The option 11y would result from addition, so it is incorrect. Exam tip: Add or subtract coefficients only when the terms are like terms.

Substitute 3 for x: \(2x+5=2(3)+5=6+5=11\). Therefore, the correct value is 11. An answer of 8 may result from incorrectly subtracting 5. Exam tip: substitute the variable value first, then perform multiplication before addition or subtraction.

Given a = 10, substitute it into the expression a - 4: 10 - 4 = 6. Therefore, the correct answer is 6. Note that 10 is the value of a, not the value of the complete expression. Exam tip: substitute the given value for the variable before simplifying.

Substitute 4 for m: \(3m+2=3\times4+2=12+2=14\). Therefore, the correct answer is 14. The close distractor 12 results from forgetting to add \(+2\). Exam tip: after substitution, perform multiplication before addition.

Given \(t=5\), substitute 5 for \(t\): \(12-2t=12-2(5)=12-10=2\). Therefore, the correct value is 2. Note that 10 is only the value of \(2t\), not of the complete expression. Exam tip: after substitution, perform multiplication before subtraction.

“x more than 5” means add x to 5. Thus, the expression is \(5+x\), which is equal to \(x+5\) by the commutative property of addition. \(x-5\) means x less than 5. Exam tip: the phrase “more than” usually indicates addition.

Taking three times a quantity means multiplying it by 3. Therefore, three times y is 3 × y = 3y. The expression y+3 means adding 3, while y³ means multiplying y by itself three times. Exam tip: When you see “times,” identify the multiplication and coefficient.

“8 less than n” means subtracting 8 from n. Therefore, the algebraic expression is n - 8. The expression 8 - n means subtracting n from 8, so it reverses the order. Exam tip: In phrases such as “less than,” subtract the stated number from the original quantity.

In the expression 6p + 1, the term containing p is 6p. Since 6 is multiplied by p, the coefficient of p is 6. The number 1 is a constant term, so it is not the coefficient of p. Exam tip: To find a variable’s coefficient, look for the number multiplying that variable in its term.

The term containing z is -4z, which can be written as (-4) × z. Therefore, the coefficient of z is -4. The number 4 is only the magnitude; the negative sign is also part of the coefficient. Exam tip: To identify a variable’s coefficient, include the complete number multiplying it, along with its sign.

Parts of an expression separated by plus (+) or minus (−) signs are called terms. Here, the terms are \(x^2\), \(4x\), and \(4\). Therefore, the expression has 3 terms. Although \(x^2+4x+4\) can be written as a perfect square, it still has 3 terms. Exam tip: Count the parts separated by + or − signs.

Like terms have the same variables raised to the same powers; only their coefficients may differ. In 3x and 5x, the variable x has power 1 in both terms, so they are like terms. In 3x and 5y the variables differ, while in x and x^2 the powers of x differ. Exam tip: Ignore the coefficients and compare the variable parts.

Unlike terms have different variables or different powers of the same variable. In 5x and 5y, the variables are x and y, so they are unlike terms. In 7a and 2a, and in 9p and 3p, the variable and its power are the same; only the coefficients differ, so they are like terms. Exam tip: Compare variables and their powers first, not the coefficients.

In the expression \(2a+3b\), \(a\) and \(b\) are letters whose values can vary, so they are variables. Therefore, there are 2 variables. The numbers 2 and 3 are coefficients, not variables. Exam tip: Count the distinct letters whose values can change to find the number of variables.

A numerical expression contains only numbers and operations; it has no variables such as \(x\), \(a\), \(p\), or \(q\). Since \(7+5\) contains only numbers, it is a numerical expression. In contrast, \(3x+2\) contains the variable \(x\), so it is an algebraic expression. Exam tip: if an expression includes a letter representing a variable, it is not purely numerical.

\(2r-5\) is an algebraic expression because it contains the variable \(r\), the numbers 2 and 5, and subtraction. \(12+7\), \(8\div4\), and \(6\times3\) are numerical expressions because they contain no variable. Exam tip: Look for a letter representing a number to identify an algebraic expression.

There are three like terms x: \(x+x+x=(1+1+1)x=3x\). Therefore, the simplified form is \(3x\). \(x^3\) is obtained from multiplication, \(x\times x\times x\), not from addition. Exam tip: when adding like terms, add their coefficients.