Algebraic expressions

The equations are \(x+y=11\) and \(x-y=5\). Subtract the second equation from the first: \((x+y)-(x-y)=11-5\), so \(2y=6\). Hence, \(y=3\). Option 5 is the constant in \(x-y=5\), not the value of \(y\). Exam tip: While adding or subtracting simultaneous equations, carefully track the signs to eliminate the required variable.

Expand the brackets: 5(2p-3q)=10p-15q and -3(p+4q)=-3p-12q. Thus, the expression becomes 10p-15q-3p-12q+2q. Combining like terms gives (10p-3p)+(−15q−12q+2q)=7p−25q. Therefore, option A is correct. Remember that the minus sign before 3(p+4q) affects both terms inside the bracket.

Given \(m=-1\) and \(n=4\), we have \(m^2=(-1)^2=1\) and \(n^2=4^2=16\). Thus, \(3m^2n=3\times1\times4=12\), while \(-2mn^2=-2\times(-1)\times16=32\). Therefore, the value of the expression is \(12+32=44\). Option 32 is only the value of the second term, not of the complete expression. Exam tip: squaring a negative number gives a positive result.

Only like terms can be added or subtracted. The terms containing x^2y are 7x^2y and -9x^2y, so their sum is (7-9)x^2y=-2x^2y. Therefore, the coefficient of x^2y is -2. The terms 3xy^2 and 5xy^2 are not like x^2y; they add to 8xy^2 instead. Exam tip: like terms must have the same variables raised to the same powers.

First simplify the innermost bracket: \(3x-(5x-8)=3x-5x+8=-2x+8\). Then \(6x-(-2x+8)=6x+2x-8=8x-8\). Therefore, \(8x-8\) is the correct option. In \(8x+8\), the sign of the final constant is incorrect because the outer minus changes the sign of every term inside the bracket. Exam tip: When a bracket is preceded by a minus sign, change the signs of all terms inside it.

The expression has value 9 when x=3. Substituting gives \(k(3)^2-4(3)-6=9\), or \(9k-12-6=9\). Thus, \(9k-18=9\), so \(9k=27\) and \(k=3\). For example, if k=1, the expression does not evaluate to 9. Exam tip: Substitute the given value of the variable carefully into every term of the expression.

Like terms have exactly the same variables and powers. In the first option, every term has variable part (p^2q^3).

Given \(a+b=8\) and \(ab=12\), substitute these values into \(4(a+b)-ab\): \(4\times 8-12=32-12=20\). Hence, the correct answer is 20. The value 32 is only \(4(a+b)\); subtracting \(ab\) is still necessary. Exam tip: substitute the given combined expressions such as \(a+b\) and \(ab\) directly.

Apply the distributive property: \(4r(r-1)=4r^2-4r\). Also, \(-3r(2-r)=-6r+3r^2\), since \(-3r\times(-r)=+3r^2\). Combining like terms gives \(4r^2+3r^2-4r-6r=7r^2-10r\). Therefore, option A is correct. In option C, the linear terms have been combined incorrectly. Exam tip: when expanding an expression with a negative sign, check the sign of every product separately.

Combining like terms gives \(8p^2q-5p^2q=3p^2q\) and \(-3pq+6pq=3pq\). Thus, the simplified expression is \(3p^2q+3pq+4\), which has 3 terms: \(3p^2q\), \(3pq\), and \(4\). Choosing 2 would be incorrect because the constant \(4\) is also a separate term. Exam tip: Combine only terms with exactly the same variables and exponents.

In u^3v^2, the exponent of u is 3 and the exponent of v is 2, so option B is correct. In option A, the exponents are reversed: u has exponent 2 and v has exponent 3. Exam tip: read the exponent written on each variable separately.

Substituting the given values, \(x^2y=(-3)^2\times2=9\times2=18\) and \(xy^2=(-3)\times2^2=(-3)\times4=-12\). Hence, \(x^2y+xy^2=18+(-12)=6\). The distractor \(-6\) may result from handling the sign of the negative term incorrectly. Exam tip: write a negative number in brackets before squaring it; \((-3)^2=9\).

While adding polynomials, combine only like terms: \(7x^2+(-3x^2)=4x^2\), \(-4x+8x=4x\), and \(9+(-2)=7\). Therefore, the sum is \(4x^2+4x+7\). Option C incorrectly adds the linear terms. Exam tip: group terms with the same power of the variable before adding.

Given \(5x-3=22\), add 3 to both sides to get \(5x=25\), so \(x=5\). Now substitute in the expression: \(x^2+2x=5^2+2(5)=25+10=35\). Hence, 35 is the correct answer. The value 30 may result from incorrectly evaluating \(x^2\). Exam tip: first find the value of the variable from the equation, then substitute it carefully into the expression.

Like terms have exactly the same variables raised to the same powers; only their coefficients may differ. In \(-3m^2n^4\), the variable part is \(m^2n^4\), so it is a like term of \(8m^2n^4\). In \(5m^4n^2\), the powers of \(m\) and \(n\) are different, so it is not a like term. Exam tip: Ignore the coefficients and compare the variables and their exponents.

A minus sign precedes the second bracket, so the sign of every term inside it changes: \(6x^2-4xy+y^2-2x^2-5xy+3y^2\). Combining like terms gives \((6-2)x^2=4x^2\), \((-4-5)xy=-9xy\), and \((1+3)y^2=4y^2\). Hence, the simplified expression is \(4x^2-9xy+4y^2\). In option D, the sign of the \(y^2\) term is incorrect. Exam tip: when subtracting a bracket, change the sign of every term in that bracket before combining like terms.

Add the two given equations:

a-b+a+b=7+17.

The terms involving b cancel, so

2a=24.

Hence,

a=12.

Option 10 may result from incorrectly treating the result of subtracting the equations as a; subtraction actually gives 2b. Exam tip: add equations when a variable has opposite signs so that it cancels out.

The term containing x^2 is -7x^2. The numerical factor multiplying the variable part is called its coefficient, so the coefficient of x^2 is -7. Here, 4, 2, and -11 belong respectively to the x^3 term, the x term, and the constant term. Exam tip: identify the term with the required power first, then write its numerical coefficient.

Expanding the expression gives \(2(5x+3y)-7(x-y)+4x=10x+6y-7x+7y+4x\). Combining the \(x\)-terms, \(10x-7x+4x=7x\). Therefore, the coefficient of \(x\) is \(7\). The distractor \(11\) can result from mishandling the negative sign in \(-7(x-y)\). Exam tip: when a negative sign is outside parentheses, distribute it to every term inside.

In \(x^3y^4\), the powers of \(x\) and \(y\) are 3 and 4 respectively. Therefore, its total degree is \(3+4=7\), so option A is correct. \(x^2y^4\) has total degree 6; similarly, \(x^5y\) and \(xy^5\) also have total degree 6. Exam tip: for a multivariable monomial, add the exponents of all variables to find its total degree.

In a polynomial, every exponent of \(x\) must be a non-negative integer. In \(7x^4-3x+2\), the highest exponent is 4, so it is a polynomial of degree 4. Option B has highest exponent 5. Option C contains \(\sqrt{x}=x^{1/2}\), and option D contains an \(x^{-1}\) term, so neither is a polynomial. Exam tip: first check that all exponents are non-negative integers, then use the highest exponent to find the degree.

Only like terms can be combined. The terms \(9u^2v\) and \(3u^2v\) are like terms, so their sum is \(12u^2v\). Similarly, \(-5uv+8uv=3uv\), while the constant term \(-10\) remains unchanged. Therefore, the simplified expression is \(12u^2v+3uv-10\). In option A, the coefficients of the \(uv\) terms have been combined incorrectly. Exam tip: before combining terms, check that both the variables and their powers are exactly the same.

On substituting \(x=-4\), \(x^3=(-4)^3=-64\) and \(x^2=(-4)^2=16\). Hence the numerator is \(-64-2(16)+8=-64-32+8=-88\). Therefore, \(\frac{-88}{-4}=22\), so option B is correct. A value such as \(26\) can result from mishandling the negative sign in \(-2x^2\). Exam tip: the square of a negative number is positive, whereas its cube is negative.

Combine like terms while adding: \(3x^2-x^2=2x^2\), \(-2x+6x=4x\), and \(5-9=-4\). Hence, the resulting expression is \(2x^2+4x-4\). In option D, the coefficients of the \(x^2\) terms have been added incorrectly. Exam tip: add the \(x^2\), \(x\), and constant terms separately.

While subtracting the second polynomial, the sign of every term in it changes: \((12x^2-5x+4)-(7x^2-9x+10)=12x^2-5x+4-7x^2+9x-10\). Combining like terms gives \(5x^2+4x-6\). In option C, the \(-9x\) term has not been correctly changed to \(+9x\) during subtraction. Exam tip: when a minus sign precedes brackets, reverse every sign inside the brackets.