GSEB Class 7 Maths Solutions Chapter 12 Algebraic Expressions Exercise 12.3

Question 1. If m =2, find the value of:
(i) m - 2
(ii) 3m - 5
(iii) 9 - 5m
(iv) \( 3m^2 - 2m -7 \)
(v) \( \frac { 5m }{ 2 } - 4 \)
Answer:
(i) We need to find the value of \( m - 2 \). Given that \( m = 2 \), we substitute this value into the expression. So, \( m - 2 = 2 - 2 = 0 \).
(ii) We need to find the value of \( 3m - 5 \). Given that \( m = 2 \), we substitute this into the expression. So, \( 3(m) - 5 = 3(2) - 5 = 6 - 5 = 1 \).
(iii) We need to find the value of \( 9 - 5m \). Given that \( m = 2 \), we substitute this into the expression. So, \( 9 - 5(m) = 9 - 5(2) = 9 - 10 = -1 \).
(iv) We need to find the value of \( 3m^2 - 2m - 7 \). Given that \( m = 2 \), we substitute this into the expression. So, \( 3(m)^2 - 2(m) - 7 = 3(2)^2 - 2(2) - 7 = 3(4) - 4 - 7 = 12 - 4 - 7 = 8 - 7 = 1 \).
(v) We need to find the value of \( \frac { 5m }{ 2 } - 4 \). Given that \( m = 2 \), we substitute this into the expression. So, \( \frac { 5(m) }{ 2 } - 4 = \frac { 5(2) }{ 2 } - 4 = \frac { 10 }{ 2 } - 4 = 5 - 4 = 1 \).
In simple words: Replace the letter 'm' with the number 2 in each math problem, then work out the answer. Follow the order of operations carefully.

Exam Tip: Remember to substitute the value of 'm' accurately into the expression and apply the order of operations (PEMDAS/BODMAS) correctly to avoid errors, especially with exponents and multiplication.

Question 2. If p = - 2 find the value of:
(i) \( 4p + 7 \)
(ii) \( - 3p^2 + 4p + 7 \)
(iii) \( - 2p^3 - 3p^2 + 4p + 7 \)
Answer:
(i) We need to find the value of \( 4p + 7 \). Given that \( p = -2 \), we substitute this into the expression. So, \( 4(p) + 7 = 4(-2) + 7 = -8 + 7 = -1 \).
(ii) We need to find the value of \( -3p^2 + 4p + 7 \). Given that \( p = -2 \), we substitute this into the expression. So, \( -3(p)^2 + 4(p) + 7 = -3(-2)^2 + 4(-2) + 7 = -3(4) + (-8) + 7 = -12 - 8 + 7 = -20 + 7 = -13 \).
(iii) We need to find the value of \( -2p^3 - 3p^2 + 4p + 7 \). Given that \( p = -2 \), we substitute this into the expression. So, \( -2(-2)^3 - 3(-2)^2 + 4(-2) + 7 = -2(-8) - 3(4) + 4(-2) + 7 = 16 - 12 - 8 + 7 = 4 - 8 + 7 = -4 + 7 = 3 \).
In simple words: Replace the letter 'p' with -2 in each problem. Be extra careful with negative numbers and powers, as they can sometimes make calculations tricky.

Exam Tip: When substituting negative values into expressions with exponents, always use parentheses, e.g., \( (-2)^2 \) to ensure the negative sign is included in the squaring operation. This helps prevent common sign errors.

Question 3. Find the value of the following expressions, when x = - 1:
(i) \( 2x - 7 \)
(ii) \( -x + 2 \)
(iii) \( x^2 + 2x + 1 \)
(iv) \( 2x^2 - x - 2 \)
Answer:
(i) We need to find the value of \( 2x - 7 \). Given that \( x = -1 \), we substitute this into the expression. So, \( 2(x) - 7 = 2(-1) - 7 = -2 - 7 = -9 \).
(ii) We need to find the value of \( -x + 2 \). Given that \( x = -1 \), we substitute this into the expression. So, \( -(x) + 2 = -(-1) + 2 = 1 + 2 = 3 \).
(iii) We need to find the value of \( x^2 + 2x + 1 \). Given that \( x = -1 \), we substitute this into the expression. So, \( (x)^2 + 2(x) + 1 = (-1)^2 + 2(-1) + 1 = 1 - 2 + 1 = 0 \).
(iv) We need to find the value of \( 2x^2 - x - 2 \). Given that \( x = -1 \), we substitute this into the expression. So, \( 2(x)^2 - x - 2 = 2(-1)^2 - (-1) - 2 = 2(1) + 1 - 2 = 2 + 1 - 2 = 3 - 2 = 1 \).
In simple words: For each expression, put -1 in place of 'x'. Then, do the math steps carefully to get the final number. Remember how negative numbers work with signs.

Exam Tip: Pay close attention to the signs when substituting negative numbers. A negative times a negative equals a positive, and a negative times a positive equals a negative. This is a common source of mistakes.

Question 4. If a = 2, b = - 2, find the value of:
(i) \( a^2 + b^2 \)
(ii) \( a^2 + ab + b^2 \)
(iii) \( a^2 - b^2 \)
Answer:
(i) We need to find the value of \( a^2 + b^2 \). Given that \( a = 2 \) and \( b = -2 \), we substitute these into the expression. So, \( (a)^2 + (b)^2 = (2)^2 + (-2)^2 = 4 + 4 = 8 \).
(ii) We need to find the value of \( a^2 + ab + b^2 \). Given that \( a = 2 \) and \( b = -2 \), we substitute these into the expression. So, \( (a)^2 + (a)(b) + (b)^2 = (2)^2 + (2)(-2) + (-2)^2 = 4 - 4 + 4 = 4 \).
(iii) We need to find the value of \( a^2 - b^2 \). Given that \( a = 2 \) and \( b = -2 \), we substitute these into the expression. So, \( (a)^2 - (b)^2 = (2)^2 - (-2)^2 = 4 - 4 = 0 \).
In simple words: Put 2 where you see 'a' and -2 where you see 'b'. Then, solve each problem, making sure to handle the positive and negative numbers correctly, especially with exponents.

Exam Tip: Remember that \( (-2)^2 \) is \( (-2) \times (-2) = 4 \), not \( -4 \). Carefully manage negative signs when squaring or multiplying terms, as this is a frequent error point.

Question 5. When a = 0, b = - 1, find the value of the given expressions:
(i) \( 2a + 2b \)
(ii) \( 2a^2 + b^2 + 1 \)
(iii) \( 2a^2b + 2ab^2 + ab \)
(iv) \( a^2 + ab + 2 \)
Answer:
(i) We need to find the value of \( 2a + 2b \). Given that \( a = 0 \) and \( b = -1 \), we substitute these into the expression. So, \( 2(a) + 2(b) = 2(0) + 2(-1) = 0 - 2 = -2 \).
(ii) We need to find the value of \( 2a^2 + b^2 + 1 \). Given that \( a = 0 \) and \( b = -1 \), we substitute these into the expression. So, \( 2(a)^2 + (b)^2 + 1 = 2(0)^2 + (-1)^2 + 1 = 2(0) + 1 + 1 = 0 + 1 + 1 = 2 \).
(iii) We need to find the value of \( 2a^2b + 2ab^2 + ab \). Given that \( a = 0 \) and \( b = -1 \), we substitute these into the expression. So, \( 2(a)^2(b) + 2(a)(b)^2 + (a)(b) = 2(0)^2(-1) + 2(0)(-1)^2 + (0)(-1) = 2(0)(-1) + 2(0)(1) + (0)(-1) = 0 + 0 + 0 = 0 \).
(iv) We need to find the value of \( a^2 + ab + 2 \). Given that \( a = 0 \) and \( b = -1 \), we substitute these into the expression. So, \( (a)^2 + (a)(b) + 2 = (0)^2 + (0)(-1) + 2 = 0 + 0 + 2 = 2 \).
In simple words: Replace 'a' with 0 and 'b' with -1 in each problem. When multiplying by 0, the result is always 0. Be careful with negative signs when you multiply by -1.

Exam Tip: Zero in multiplication makes the whole term zero, which can simplify calculations. Always remember that \( 0 \times \text{anything} = 0 \). This can help you quickly evaluate expressions.

Question 6. Simplify the expressions and find the value if x is equal to 2.
(i) \( x + 7 + 4(x - 5) \)
(ii) \( 3(x + 2) + 5x-7 \)
(iii) \( 6x + 5(x - 2) \)
(iv) \( 4(2x - 1) + 3x + 11 \)
Answer:
(i) We have the expression \( x + 7 + 4(x - 5) \). First, we simplify the expression. Distribute the 4: \( x + 7 + 4x - 20 \). Combine like terms: \( (x + 4x) + (7 - 20) = 5x - 13 \). Now, substitute \( x = 2 \): \( 5(2) - 13 = 10 - 13 = -3 \).
(ii) We have the expression \( 3(x + 2) + 5x - 7 \). First, we simplify the expression. Distribute the 3: \( 3x + 6 + 5x - 7 \). Combine like terms: \( (3x + 5x) + (6 - 7) = 8x - 1 \). Now, substitute \( x = 2 \): \( 8(2) - 1 = 16 - 1 = 15 \).
(iii) We have the expression \( 6x + 5(x - 2) \). First, we simplify the expression. Distribute the 5: \( 6x + 5x - 10 \). Combine like terms: \( (6x + 5x) - 10 = 11x - 10 \). Now, substitute \( x = 2 \): \( 11(2) - 10 = 22 - 10 = 12 \).
(iv) We have the expression \( 4(2x - 1) + 3x + 11 \). First, we simplify the expression. Distribute the 4: \( 8x - 4 + 3x + 11 \). Combine like terms: \( (8x + 3x) + (-4 + 11) = 11x + 7 \). Now, substitute \( x = 2 \): \( 11(2) + 7 = 22 + 7 = 29 \).
In simple words: First, make each math problem simpler by combining similar parts and getting rid of brackets. Then, put the number 2 in place of 'x' and work out the final answer for each simplified problem.

Exam Tip: Always simplify the expression completely before substituting the value of the variable. This reduces the number of calculations and decreases the chances of making errors.

Question 7. Simplify these expressions and find their values if x = 3, a = -1, b = -2.
(i) \( 3x - 5 - x + 9 \)
(ii) \( 2 - 8x + 4x + 4 \)
(iii) \( 3a + 5 - 8a + 1 \)
(iv) \( 10 - 3b - 4 - 5b \)
(v) \( 2a - 2b - 4 - 5 + a \)
Answer:
(i) We have the expression \( 3x - 5 - x + 9 \). First, we simplify by combining like terms: \( (3x - x) + (-5 + 9) = 2x + 4 \). Now, substitute \( x = 3 \): \( 2(3) + 4 = 6 + 4 = 10 \).
(ii) We have the expression \( 2 - 8x + 4x + 4 \). First, we simplify by combining like terms: \( (-8x + 4x) + (2 + 4) = -4x + 6 \). Now, substitute \( x = 3 \): \( -4(3) + 6 = -12 + 6 = -6 \).
(iii) We have the expression \( 3a + 5 - 8a + 1 \). First, we simplify by combining like terms: \( (3a - 8a) + (5 + 1) = -5a + 6 \). Now, substitute \( a = -1 \): \( -5(-1) + 6 = 5 + 6 = 11 \).
(iv) We have the expression \( 10 - 3b - 4 - 5b \). First, we simplify by combining like terms: \( (10 - 4) + (-3b - 5b) = 6 - 8b \). Now, substitute \( b = -2 \): \( 6 - 8(-2) = 6 + 16 = 22 \).
(v) We have the expression \( 2a - 2b - 4 - 5 + a \). First, we simplify by combining like terms: \( (2a + a) - 2b + (-4 - 5) = 3a - 2b - 9 \). Now, substitute \( a = -1 \) and \( b = -2 \): \( 3(-1) - 2(-2) - 9 = -3 + 4 - 9 = 1 - 9 = -8 \).
In simple words: For each problem, first, make it simpler by grouping similar terms together. Then, put in the given numbers for 'x', 'a', and 'b', and work out the answer. Be extra careful with negative signs.

Exam Tip: When simplifying expressions with multiple variables and constants, ensure you group the terms correctly (e.g., all 'x' terms together, all 'a' terms together, and all constant terms together) before performing substitution.

Question 8.
(i) If z = 10,find the value of \( z^3 - 3(z - 10) \).
(ii) If p = - 10, find the value of \( p^2 - 2p - 100 \).
Answer:
(i) We need to find the value of \( z^3 - 3(z - 10) \) when \( z = 10 \). Substitute \( z = 10 \) into the expression: \( (10)^3 - 3(10 - 10) = 1000 - 3(0) = 1000 - 0 = 1000 \).
(ii) We need to find the value of \( p^2 - 2p - 100 \) when \( p = -10 \). Substitute \( p = -10 \) into the expression: \( (-10)^2 - 2(-10) - 100 = 100 + 20 - 100 = 120 - 100 = 20 \).
In simple words: For the first part, replace 'z' with 10. For the second part, replace 'p' with -10. Then, do the calculations, remembering that a negative number squared becomes positive.

Exam Tip: Always handle parentheses first, especially when there's a subtraction inside, like \( (z - 10) \). Also, remember that squaring a negative number results in a positive number.

Question 9. What should be the value of a if the value of \( 2x^2 + x - a \) equals to 5, when x = 0?
Answer: We are given the expression \( 2x^2 + x - a \) and told its value is 5 when \( x = 0 \). Substitute \( x = 0 \) into the expression: \( 2(0)^2 + (0) - a = 5 \). This simplifies to \( 0 + 0 - a = 5 \), which means \( -a = 5 \). To find 'a', we multiply both sides by -1, giving us \( a = -5 \). Therefore, the required value of 'a' is -5.
In simple words: Put 0 in place of 'x' in the math problem. The problem then becomes '0 minus a equals 5'. This means 'a' must be -5 to make the equation true.

Exam Tip: When given an equation and values for some variables, substitute those values first to simplify the equation. Then, solve for the unknown variable using standard algebraic techniques.

Question 10. Simplify the expression and find its value when a = 5 and b = - 3.
\( 2(a^2 + ab) + 3 - ab \)
Answer: We have the expression \( 2(a^2 + ab) + 3 - ab \). First, we simplify the expression. Distribute the 2: \( 2a^2 + 2ab + 3 - ab \). Combine like terms: \( 2a^2 + (2ab - ab) + 3 = 2a^2 + ab + 3 \). Now, substitute \( a = 5 \) and \( b = -3 \): \( 2(5)^2 + (5)(-3) + 3 = 2(25) + (-15) + 3 = 50 - 15 + 3 = 35 + 3 = 38 \).
In simple words: First, make the problem simpler by getting rid of brackets and grouping similar parts. Then, put 5 for 'a' and -3 for 'b' into the simplified problem, and work out the final number.

Exam Tip: Always distribute numbers into parentheses correctly and combine like terms before substituting variable values. This methodical approach ensures accuracy, especially with multiple variables and operations.