Get authentic, step-by-step DAV Class 6 Maths Chapter 6 Solutions: Introduction to Algebra from the official DAV Secondary Mathematics textbook. In this comprehensive guide, every worksheet from Worksheet 1 to Worksheet 7 and Brain Teasers is solved with detailed explanations, substitution steps, and algebraic rules.
Chapter Overview: Introduction to Algebra
Algebra is a branch of mathematics where letters and symbols (called variables or literal numbers) represent numbers. This chapter covers writing algebraic expressions, evaluating expressions, identifying coefficients, like/unlike terms, addition, subtraction, and perimeter word problems.
Worksheet 1: Framing Algebraic Expressions
1. Write the algebraic expression for each of the following statements:
(a) 7 added to \(x\)
Solution: \(x + 7\)
(b) \(y\) decreased by 9
Solution: \(y – 9\)
(c) The product of \(p\) and \(q\)
Solution: \(pq\)
(d) One-fourth of a number \(z\)
Solution: \( rac{z}{4}\) or \( rac{1}{4}z\)
(e) 5 times \(m\) added to 3 times \(n\)
Solution: \(5m + 3n\)
(f) 8 subtracted from the quotient of \(x\) by \(y\)
Solution: \( rac{x}{y} – 8\)
Worksheet 2: Evaluation of Algebraic Expressions
1. If \(a = 3\), \(b = 2\), and \(c = -1\), evaluate the following:
(a) \(2a + 3b\)
Solution:
\(2(3) + 3(2) = 6 + 6 = 12\)
(b) \(4a – 5b + 2c\)
Solution:
\(4(3) – 5(2) + 2(-1) = 12 – 10 – 2 = 0\)
(c) \(a^2 + b^2 – c^2\)
Solution:
\((3)^2 + (2)^2 – (-1)^2 = 9 + 4 – 1 = 12\)
Worksheet 3: Terms, Coefficients & Like/Unlike Terms
1. Identify the numerical coefficient of each term:
(a) \(-7xy\) âž” Numerical Coefficient: \(-7\)
(b) \( rac{3}{4}a^2b\) âž” Numerical Coefficient: \( rac{3}{4}\)
(c) \(-pqr\) âž” Numerical Coefficient: \(-1\)
2. Group the like terms from the following set:
Set: \(3ab, -5a^2, 7ba, 12b^2, -4ab, 9a^2\)
Solution:
• Group 1 (\(ab\) terms): \(3ab, 7ba, -4ab\)
• Group 2 (\(a^2\) terms): \(-5a^2, 9a^2\)
• Group 3 (\(b^2\) terms): \(12b^2\)
Worksheet 4 & 5: Addition of Algebraic Expressions
1. Add the following expressions:
\((3x + 5y – 4z)\) and \((2x – 3y + 7z)\)
Solution:
Using column method:
\(egin{aligned}
&\phantom{+}3x + 5y – 4z \
&+ 2x – 3y + 7z \
\hline
&= (3+2)x + (5-3)y + (-4+7)z \
&= 5x + 2y + 3z
\end{aligned}\)
2. Add: \((4a^2 – 3ab + 2b^2)\), \((-2a^2 + 5ab – b^2)\), and \((a^2 – ab + 3b^2)\)
Solution:
\(egin{aligned}
&= (4 – 2 + 1)a^2 + (-3 + 5 – 1)ab + (2 – 1 + 3)b^2 \
&= 3a^2 + ab + 4b^2
\end{aligned}\)
Worksheet 6: Subtraction of Algebraic Expressions
1. Subtract \((2a – 5b + 3c)\) from \((5a + 2b – c)\):
Solution:
\(egin{aligned}
&= (5a + 2b – c) – (2a – 5b + 3c) \
&= 5a + 2b – c – 2a + 5b – 3c \
&= (5 – 2)a + (2 + 5)b + (-1 – 3)c \
&= 3a + 7b – 4c
\end{aligned}\)
2. What should be added to \((x^2 + xy + y^2)\) to obtain \((2x^2 + 3xy)\)?
Solution:
Required expression = \((2x^2 + 3xy) – (x^2 + xy + y^2)\)
\(= 2x^2 + 3xy – x^2 – xy – y^2\)
\(= x^2 + 2xy – y^2\)
Worksheet 7: Word Problems on Perimeter and Real Life
1. The three sides of a triangle are \((2x + 3)\) cm, \((3x – 1)\) cm, and \((4x + 5)\) cm. Find the perimeter of the triangle.
Solution:
Perimeter of a triangle = Sum of all three sides
\(= (2x + 3) + (3x – 1) + (4x + 5)\)
\(= (2 + 3 + 4)x + (3 – 1 + 5)\)
\(= (9x + 7) ext{ cm}\)
2. A rectangle has length \((3a + 2b)\) m and breadth \((2a – b)\) m. Find its perimeter.
Solution:
Perimeter of rectangle = \(2 imes ( ext{Length} + ext{Breadth})\)
\(= 2 imes [(3a + 2b) + (2a – b)]\)
\(= 2 imes [5a + b]\)
\(= 10a + 2b ext{ metres}\)
Brain Teasers Solutions
Q1. Multiple Choice Questions (Tick the correct option):
(a) The value of \(2x^3 – 3x^2 + 5x – 7\) when \(x = -1\) is:
Answer: \(2(-1)^3 – 3(-1)^2 + 5(-1) – 7 = -2 – 3 – 5 – 7 = -17\).
(b) The degree of the algebraic expression \(4x^2y^3 + 7x^3y – 2xy\) is:
Answer: Degree = Highest sum of exponents in a term = \(2 + 3 = 5\).
(c) How much is \(-2x + 5\) smaller than \(3x – 2\)?
Answer: \((3x – 2) – (-2x + 5) = 3x – 2 + 2x – 5 = 5x – 7\).
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