Algebraic expressions

The number 2 more than \(x\) is \(x+2\), and the number 3 less than \(x\) is \(x-3\). Therefore, their sum is \((x+2)+(x-3)=2x-1\). The expression \(2x+5\) would result from adding 3 instead of subtracting it. Exam tip: translate “more than” as addition and “less than” as subtraction before simplifying.

(\frac{x}{3}=\frac{2x}{6}), so (\frac{5x}{6}-\frac{2x}{6}=\frac{x}{2}). Make common denominators for fraction terms.

Given \(x+y=5\) and \(x-y=1\), substitute these values directly: \(2(x+y)+3(x-y)=2\times5+3\times1=10+3=13\). Therefore, 13 is correct. The value 11 does not result from correct substitution. Exam tip: When \(x+y\) and \(x-y\) are given directly, there is no need to find \(x\) and \(y\) separately.

Combining (x^2), (xy), and (y^2) terms separately gives (3x^2+5xy+y^2). Do not mix different variable parts.

Use the distributive law from the innermost bracket outward. First, 2(4 - x) = 8 - 2x. Thus the square-bracket expression is 3x - (8 - 2x), which becomes 3x - 8 + 2x = 5x - 8. The complete expression is then 10 - (5x - 8). The minus sign before the bracket changes both signs inside, so 10 - 5x + 8 = 18 - 5x. Therefore, option B is correct. Option A would follow if the 3x term were combined incorrectly, option C reverses the effect of the outer subtraction, and option D ignores the coefficient of x created by combining like terms.

Substituting t=2 gives 3(2^4)-2(2^3)+2. Since 2^4=16 and 2^3=8, the value is 3×16-2×8+2=48-16+2=34. The value 36 may result from incorrectly omitting the final +t term. Exam tip: evaluate powers first, then perform multiplication and addition/subtraction.

The difference is (a-b), and three times it is (3(a-b)). Adding the sum (a+b) gives (3(a-b)+(a+b)).

A minus sign occurs before the last bracket, so the signs of both terms inside it must change: \((2x+3)+(4x-7)-(x+5)=2x+3+4x-7-x-5\). Combining like terms gives \((2x+4x-x)=5x\) and \((3-7-5)=-9\). Therefore, the simplified form is \(5x-9\). The option \(5x+1\) results from handling the constant \(-5\) incorrectly. Exam tip: whenever a bracket is preceded by a minus sign, reverse the sign of every term inside it.

\(7x^2\), \(-3x\), and \(5\) are three separate terms, so the expression is a trinomial. For degree, take the greatest exponent of the variable: \(x^2\) has exponent 2. The coefficient 3 in \(-3x\) is not the degree. Exam tip: count terms and find degree separately.

The subtraction is \((4x-2y)-(x+5y)\). The minus sign before the second bracket changes the signs of both its terms: \(4x-2y-x-5y\). Combining like terms gives \(4x-x=3x\) and \(-2y-5y=-7y\), so the result is \(3x-7y\). In \(5x-7y\), the \(x\)-terms have been incorrectly added. Exam tip: when a bracket is preceded by a minus sign, change the sign of every term inside it.

\(3x^2y\) and \(7x^2y\) are like terms, so adding their coefficients gives \(10x^2y\). Similarly, \(-5xy^2\) and \(-2xy^2\) combine to give \(-7xy^2\). Therefore, the simplified expression is \(10x^2y-7xy^2\). Option B incorrectly treats the negative second terms as positive. Exam tip: combine only those terms that have the same variables raised to the same powers.

On substituting \(q=3\), we get \(q^3=3^3=27\). Hence, \(\frac{q^3-q}{3}=\frac{27-3}{3}=\frac{24}{3}=8\). Therefore, option A is correct. The value 6 can result from an error in evaluating the exponent or simplifying the numerator. Exam tip: evaluate powers first, then simplify the numerator before dividing.

The four consecutive integers are \(n, n+1, n+2\), and \(n+3\). Their sum is \(n+(n+1)+(n+2)+(n+3)=4n+6\), so option A is correct. In \(4n+4\), the extra parts of the consecutive integers, \(1+2+3=6\), have not been added correctly. Exam tip: Write consecutive integers by adding \(1,2,3\), and so on, to the first integer.

First simplify the square bracket: \(x-2(1-x)=x-2+2x=3x-2\). Also, \(2(3x-4)=6x-8\). Therefore, the expression becomes \(6x-8-(3x-2)=6x-8-3x+2=3x-6\). Hence, option A is correct. The result \(3x-10\) can arise if the sign of \(+2\) is handled incorrectly while subtracting \((3x-2)\). Exam tip: when a minus sign occurs before a bracket, change the sign of every term inside it.

Substituting the given values, x+y=2+1=3. Therefore, 2(x+y)^2-3xy=2(3)^2-3(2)(1)=2×9-6=18-6=12. Hence, the correct answer is 12. The value 18 is only the first term, 2(x+y)^2; the term 3xy must also be subtracted. Exam tip: evaluate brackets first, then powers and multiplication, and finally addition or subtraction.

On substituting the given values, x+y=2+1=3 and xy=2×1=2. Therefore, 2(x+y)^2-3xy=2(3)^2-3(2)=2×9-6=12. Option C, 18, is only the value of 2(3)^2; it does not subtract 3xy. Exam tip: evaluate brackets and powers before carrying out multiplication and subtraction.

Combine like terms in the expression: \(2a+5a-4a=3a\) and \(-3b+7b=4b\). Hence, the simplified form is \(3a+4b\). In \(11a+4b\), the negative sign of \(-4a\) has been ignored while combining the \(a\)-terms. Exam tip: add or subtract only terms having the same variable and exponent.

The perimeter of a rectangle is \(2(\text{length}+\text{breadth})\). Thus, \(2[(3x-2)+(2x+5)]=2(5x+3)=10x+6\). Hence, the correct expression is \(10x+6\). The expression \(5x+3\) is only the sum of the length and breadth, not the perimeter. Exam tip: A perimeter includes all four sides, so multiply the sum of length and breadth by 2.

Substitute x=1. Since \(1^3=1\) and \(1^2=1\), \(4(1)^3-6(1)^2+5(1)-7=4-6+5-7=-4\). Therefore, -4 is correct. The nearby distractor -3 can result from an error in addition or subtraction. Exam tip: after substitution, write the sign of every term before simplifying.

First simplify the innermost bracket: 3x - (x + 4) = 3x - x - 4 = 2x - 4. Then, 5x - 2(2x - 4) = 5x - 4x + 8 = x + 8. Hence, x + 8 is correct. The option x - 8 results from handling the sign incorrectly when multiplying 2 by -4. Exam tip: When a bracket is preceded by a minus sign or a multiplier, apply it to every term inside the bracket.