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Finding the Unknown - Solving Equations Systematically

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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An equation is a mathematical statement where two expressions are equal, denoted by the '=' sign. The expression on the left is the LHS (Left Hand Side) and the expression on the right is the RHS (Right Hand Side).

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To solve an equation systematically, we must perform the same mathematical operation on both sides of the equation to maintain the balance.

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The systematic method involves: adding the same number to both sides, subtracting the same number from both sides, multiplying both sides by the same non-zero number, or dividing both sides by the same non-zero number.

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The Transposition Method is a shortcut where a term is moved from one side of the equation to the other by changing its sign: addition becomes subtraction (++ to −-), subtraction becomes addition (−- to ++), multiplication becomes division (×\times to ÷\div), and division becomes multiplication (÷\div to ×\times).

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A solution is the value of the variable that makes the LHS=RHSLHS = RHS. We can verify the solution by substituting the value back into the original equation.

📐Formulae

ax+b=cax + b = c

x+a=b  ⟹  x=b−ax + a = b \implies x = b - a

x−a=b  ⟹  x=b+ax - a = b \implies x = b + a

ax=b  ⟹  x=baax = b \implies x = \frac{b}{a}

xa=b  ⟹  x=b×a\frac{x}{a} = b \implies x = b \times a

💡Examples

Problem 1:

Solve the equation systematically: x+8=15x + 8 = 15

Solution:

Step 1: Subtract 88 from both sides: x+8−8=15−8x + 8 - 8 = 15 - 8 Step 2: Simplify: x=7x = 7

Explanation:

To isolate xx, we perform the inverse operation of adding 88, which is subtracting 88 from both sides of the equation.

Problem 2:

Solve for yy using the transposition method: 4y−7=214y - 7 = 21

Solution:

Step 1: Transpose −7-7 to the RHS: 4y=21+74y = 21 + 7 4y=284y = 28 Step 2: Transpose 44 (multiplier) to the RHS as a divisor: y=284y = \frac{28}{4} y=7y = 7

Explanation:

We first move the constant term by changing its sign from negative to positive. Then, we move the coefficient of yy by changing multiplication to division.

Problem 3:

Solve the equation: z3+5=2\frac{z}{3} + 5 = 2

Solution:

Step 1: Subtract 55 from both sides: z3=2−5\frac{z}{3} = 2 - 5 z3=−3\frac{z}{3} = -3 Step 2: Multiply both sides by 33: z=−3×3z = -3 \times 3 z=−9z = -9

Explanation:

First, we eliminate the constant term +5+5 by subtracting it from the RHS. Then, we eliminate the denominator 33 by multiplying it with the RHS.