Algebraic Expressions

Q1: Multiple Choice Type

i. The value of \(x – \overline{x – y}\) is:

Step 1: Solve the inner bracket:
\(\overline{x – y} = x – y\)
Step 2: Subtract:
\(x – (x – y) = x – x + y = y\)
Answer: c. \(y\)

ii. \((5x – 4y) – (5y – 4x)\) is equal to:

Step 1: Remove brackets:
\(5x – 4y – 5y + 4x\)
Step 2: Combine like terms:
\((5x + 4x) + (-4y – 5y) = 9x – 9y = 9(x – y)\)
Answer: a. \(9(x – y)\)

iii. \(x(y – x – \overline{x – y})\) is equal to:

Step 1: Solve the bar first:
\(\overline{x – y} = x – y\)
Step 2: Substitute in the expression:
\(x(y – x – (x – y)) = x(y – x – x + y)\)
Step 3: Simplify:
\(x(2y – 2x) = 2x(y – x)\)
Answer: b. \(2x(y – x)\)

iv. \((2x – y) + \overline{2x – y}\) is equal to:

Step 1: Evaluate the bar:
\(\overline{2x – y} = 2x – y\)
Step 2: Add:
\((2x – y) + (2x – y) = 4x – 2y\)
Answer: d. \(4x – 2y\)

v. \(x(y – z) + y(z – x) – z(y – x)\) is equal to:

Step 1: Expand each term:
\(x(y – z) = xy – xz\)
\(y(z – x) = yz – xy\)
\(-z(y – x) = -zy + zx\)
Step 2: Combine all terms:
\((xy – xz) + (yz – xy) + (-zy + zx)\)
Step 3: Cancel terms:
+xy and -xy cancel
-zy and +yz cancel
-xz and +zx cancel
Everything cancels out → 0
Answer: a. 0

Q2: Simplify \(a^2 – 2a + \left\{5a^2 – \left(3a – 4a^2\right)\right\}\)

Given Expression: \[ a^2 – 2a + \left\{5a^2 – \left(3a – 4a^2\right)\right\} \]Step 1: Simplify the inner bracket: \[ 3a – 4a^2 \]Step 2: Substitute and simplify inside the curly braces: \[ 5a^2 – (3a – 4a^2) = 5a^2 – 3a + 4a^2 = (5a^2 + 4a^2) – 3a = 9a^2 – 3a \]Step 3: Now the expression becomes: \[ a^2 – 2a + 9a^2 – 3a \]Step 4: Combine like terms: \[ (a^2 + 9a^2) + (-2a – 3a) = 10a^2 – 5a \]Answer: \(10a^2 – 5a\)

Q3: Simplify \(x – y – \left\{ x – y – (x + y) – \overline{x – y} \right\}\)

Given Expression: \[ x – y – \left\{ x – y – (x + y) – \overline{x – y} \right\} \]Step 1: Simplify the expression inside the curly braces: \[ x – y – (x + y) – \overline{x – y} \]Step 2: First simplify \(\overline{x – y}\) (negation of \(x – y\)): \[ \overline{x – y} = (x – y) \]Step 3: Substitute and simplify inside the braces: \[ x – y – (x + y) – (x – y) = x – y – x – y – x + y \]Break it down: \[ (x – x – x) + (-y – y + y) = -x + (-y) \]Simplify: \[ -x – y \]Step 4: Now the entire expression becomes: \[ x – y – (-x – y) \]Step 5: Simplify: \[ x – y + x + y = (x + x) + (-y + y) = 2x + 0 = 2x \]Answer: \(2x\)

Q4: Simplify \(-3(1 – x^2) – 2 \left\{ x^2 – (3 – 2x^2) \right\}\)

Given Expression: \[ -3(1 – x^2) – 2 \left\{ x^2 – (3 – 2x^2) \right\} \]Step 1: Simplify the inner bracket: \[ 3 – 2x^2 \]Step 2: Substitute and simplify inside the curly braces: \[ x^2 – (3 – 2x^2) = x^2 – 3 + 2x^2 = (x^2 + 2x^2) – 3 = 3x^2 – 3 \]Step 3: Now the expression becomes: \[ -3(1 – x^2) – 2(3x^2 – 3) \]Step 4: Expand both terms: \[ -3 \times 1 + (-3) \times (-x^2) – 2 \times 3x^2 + (-2) \times (-3) \]Calculate: \[ -3 + 3x^2 – 6x^2 + 6 \]Step 5: Combine like terms: \[ (-3 + 6) + (3x^2 – 6x^2) = 3 – 3x^2 \]Answer: \(3 – 3x^2\)

Q5: Simplify \(2\left\{ m – 3\left( n + \overline{m – 2n} \right) \right\}\)

Given Expression: \[ 2 \left\{ m – 3 \left( n + \overline{m – 2n} \right) \right\} \]Step 1: Simplify the negation \(\overline{m – 2n}\): \[ \overline{m – 2n} = m – 2n \]Step 2: Simplify inside the parenthesis: \[ n + (m – 2n) = n + m – 2n = m – n \]Step 3: Multiply by \(-3\): \[ -3 \times (m – n) = -3m + 3n \]Step 4: Now simplify inside the braces: \[ m + (-3m + 3n) = -2m + 3n \]Step 5: Multiply entire expression by 2: \[ 2 \times (-2m + 3n) = -4m + 6n \]Answer: \(-4m + 6n\)

Q6: Simplify \(3x – \left[ 3x – \left\{ 3x – \left( 3x – \overline{3x – y} \right) \right\} \right]\)

Given Expression: \[ 3x – \left[ 3x – \left\{ 3x – \left( 3x – \overline{3x – y} \right) \right\} \right] \]Step 1: Simplify the negation \(\overline{3x – y}\): \[ \overline{3x – y} = 3x – y \]Step 2: Simplify inside the innermost parentheses: \[ 3x – \overline{3x – y} = 3x – (3x – y) = 3x – 3x + y = y \]Step 3: Simplify inside the curly braces: \[ 3x – (y) = 3x – y = 3x – y \]Step 4: Simplify inside the square brackets: \[ 3x – (3x – y) = 3x – 3x + y = y \]Step 5: Finally simplify the entire expression: \[ 3x – (y) = 3x – y = 3x – y \]Answer: \(3x – y\)

Q7: Simplify \(p^2x – 2 \left\{ px – 3x \left( x^2 – \overline{3a – x^2} \right) \right\}\)

Given Expression: \[ p^2x – 2 \left\{ px – 3x \left( x^2 – \overline{3a – x^2} \right) \right\} \]Step 1: Simplify the negation \(\overline{3a – x^2}\): \[ \overline{3a – x^2} = 3a – x^2 \]Step 2: Simplify inside the parenthesis: \[ x^2 – \overline{3a – x^2} = x^2 – (3a – x^2) = x^2 – 3a + x^2 = -3a + 2x^2 \]Step 3: Multiply inside the braces: \[ 3x \times (-3a + 2x^2) = -9ax + 6x^3 \]Step 4: Simplify inside the braces: \[ px – (-9ax + 6x^2) = px + 9ax + 6x^3 \]Step 5: Multiply entire braces by \(2\): \[ 2 \times (px + 9ax + 6x^3p) = 2px + 18ax + 12x^3 \]Step 6: Final expression: \[ p^2x – (2px + 18ax + 12x^3p) = p^2x – 2px – 18ax – 12x^3 \]Answer: \(p^2x – 2px – 18ax – 12x^3\)

Q8: Simplify \(2 \left[ 6 + 4 \left\{ m – 6 \left( 7 – \overline{n + p} \right) + q \right\} \right]\)

Given Expression: \[ 2 \left[ 6 + 4 \left\{ m – 6 \left( 7 – \overline{n + p} \right) + q \right\} \right] \]Step 1: Simplify the negation \(\overline{n + p}\): \[ \overline{n + p} = n + p \]Step 2: Simplify inside the parentheses: \[ 7 – \overline{n + p} = 7 – (n + p) = 7 – n – p \]Step 3: Multiply by 6: \[ 6 \times (7 – n – p) = 42 – 6n – 6p \]Step 4: Simplify inside the braces: \[ m – (42 – 6n – 6p) + q = m – 42 + 6n + 6p + q \]Step 5: Multiply by 4: \[ 4 \times (m – 42 + 6n + 6p + q) = 4m – 168 + 24n + 24p + 4q \]Step 6: Add 6: \[ 6 + (4m – 168 + 24n + 24p + 4q) = 4m – 162 + 24n + 24p + 4q \]Step 7: Multiply entire expression by 2: \[ 2 \times (4m – 162 + 24n + 24p + 4q) = 8m – 324 + 48n + 48p + 8q \]Answer: \(8m – 324 + 48n + 48p + 8q\)

Q9: Simplify \(a – \left[ a – \overline{b + a} – \left\{ a – \left( a – \overline{b – a} \right) \right\} \right]\)

Given Expression: \[ a – \left[ a – \overline{b + a} – \left\{ a – \left( a – \overline{b – a} \right) \right\} \right] \]Step 1: Simplify the negations: \[ \overline{b + a} = b + a \\ \overline{b – a} = b – a \]Step 2: Simplify the innermost bracket: \[ a – \overline{b – a} = a – (b – a) = a – b + a = 2a – b \]Step 3: Simplify the next bracket: \[ a – \left( a – \overline{b – a} \right) = a – (2a – b) = a – 2a + b = -a + b \]Step 4: Substitute back: \[ a – \left[ a – (b + a) – (-a – b) \right] = a – \left[ a – b – a – (-a + b) \right] \]Step 5: Simplify inside the square brackets: \[ a – b – a – (-a + b) = a – b – a + a – b = a – b – b = a – 2b \]Step 6: Now simplify the entire expression: \[ a – (a – 2b) = a – a + 2b = 2b \]Answer: \(2b\)

Q10: Simplify \(3x – \left[4x – \overline{3x – 5y} – 3 \left\{ 2x – \left(3x – \overline{2x – 3y}\right) \right\} \right]\)

Given Expression: \[ 3x – \left[ 4x – \overline{3x – 5y} – 3 \left\{ 2x – \left( 3x – \overline{2x – 3y} \right) \right\} \right] \]Step 1: Simplify the negations: \[ \overline{3x – 5y} = 3x – 5y \\ \overline{2x – 3y} = 2x – 3y \]Step 2: Simplify inside the innermost parentheses: \[ 3x – \overline{2x – 3y} = 3x – (2x – 3y) = 3x – 2x + 3y = x + 3y \]Step 3: Simplify inside the curly braces: \[ 2x – (x + 3y) = 2x – x – 3y = x – 3y \]Step 4: Multiply by 3: \[ 3 \times (x – 3y) = 3x – 9y \]Step 5: Simplify inside the square brackets: \[ 4x – (3x – 5y) – (3x – 9y) = 4x – 3x + 5y – 3x + 9y = (4x – 3x – 3x) + (5y + 9y) = -2x + 14y \]Step 6: Now simplify the whole expression: \[ 3x – (-2x + 14y) = 3x + 2x – 14y = 5x – 14y \]Answer: \(5x – 14y\)

Q11: Simplify \(a^5 \div a^3 + 3a \times 2a\)

Given Expression: \[ a^5 \div a^3 + 3a \times 2a \]Step 1: Simplify division of powers with same base \(a\): \[ a^5 \div a^3 = a^{5-3} = a^2 \]Step 2: Simplify multiplication: \[ 3a \times 2a = 3 \times 2 \times a \times a = 6a^2 \]Step 3: Add the two results: \[ a^2 + 6a^2 = 7a^2 \]Answer: \(7a^2\)

Q12: Simplify \(x^5 \div \left(x^2 \times y^2\right) \times y^3\)

Given Expression: \[ x^5 \div \left(x^2 y^2\right) \times y^3 \]Step 1: Simplify the division of powers with same base \(x\): \[ x^5 \div x^2 = x^{5-2} = x^3 \]Step 2: Rewrite the expression: \[ \left(x^3 \div y^2\right) \times y^3 = x^3 \times \left(\frac{y^3}{y^2}\right) \]Step 3: Simplify the powers of \(y\): \[ \frac{y^3}{y^2} = y^{3-2} = y^1 = y \]Step 4: Multiply: \[ x^3 \times y = x^3 y \]Answer: \(x^3 y\)

Q13: Simplify \(\left(x^5 \div x^2\right) \times y^2 \times y^3\)

Given Expression: \[ \left(x^5 \div x^2\right) \times y^2 \times y^3 \]Step 1: Simplify division of powers with same base \(x\): \[ x^5 \div x^2 = x^{5-2} = x^3 \]Step 2: Multiply powers of \(y\): \[ y^2 \times y^3 = y^{2+3} = y^5 \]Step 3: Multiply the results: \[ x^3 \times y^5 = x^3 y^5 \]Answer: \(x^3 y^5\)

Q14: Simplify \(\left(y^3 – 5y^2\right) \div y \times \left(y – 1\right)\)

Given Expression: \[ \left(y^3 – 5y^2\right) \div y \times \left(y – 1\right) \]Step 1: Divide each term in the numerator by \(y\): \[ \frac{y^3}{y} – \frac{5y^2}{y} = y^{3-1} – 5 y^{2-1} = y^2 – 5y \]Step 2: Multiply the result by \((y – 1)\): \[ (y^2 – 5y)(y – 1) \]Step 3: Use distributive property: \[ y^2 \times y – y^2 \times 1 – 5y \times y + 5y \times 1 = y^3 – y^2 – 5y^2 + 5y \]Step 4: Combine like terms: \[ y^3 – (y^2 + 5y^2) + 5y = y^3 – 6y^2 + 5y \]Answer: \(y^3 – 6y^2 + 5y\)