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Linear Equations in One Variable - Reducing equations to simpler form

Grade 8ICSE

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

🔑Concepts

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A linear equation in one variable is an equality involving an unknown variable with the highest power as 1. It can be visualized as a balanced weighing scale where the expression on the left-hand side (LHS) has the exact same value as the expression on the right-hand side (RHS).

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Reducing equations involves simplifying complex expressions by removing brackets using the Distributive Law. When you see a term like a(bx+c)a(bx + c), visualize the 'a' being distributed or multiplied into every term inside the parentheses to get abx+acabx + ac.

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To clear fractions in an equation, find the Least Common Multiple (LCM) of all denominators and multiply every term on both sides by this LCM. This process 'flattens' the equation, turning a multi-level fractional expression into a single-line linear format.

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Transposition is the process of moving a term from one side of the '=' sign to the other. When a term 'crosses' the equal sign, its operation changes: addition becomes subtraction, subtraction becomes addition, multiplication becomes division, and division becomes multiplication.

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Cross-multiplication is a powerful technique used when the equation is in the form of one fraction equaling another, such as ab=cd\frac{a}{b} = \frac{c}{d}. You can visualize drawing an 'X' across the equals sign, connecting the numerator of one side to the denominator of the other, resulting in ad=bcad = bc.

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Grouping like terms is the step where all terms containing the variable (like xx) are moved to the LHS and all constant numbers are moved to the RHS. Think of this as sorting different types of items into two specific bins to isolate the unknown variable.

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The final step in solving is Verification. Once you find a numerical value for the variable, substitute it back into the original LHS and RHS. If the resulting values are equal, the 'balance scale' is level, and your solution is correct.

📐Formulae

Standard Form: ax+b=cax + b = c

Distributive Property: a(b+c)=ab+aca(b + c) = ab + ac

Transposition Rule: x+a=b  ⟹  x=b−ax + a = b \implies x = b - a

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

Cross-Multiplication: ax+bcx+d=pq  ⟹  q(ax+b)=p(cx+d)\frac{ax + b}{cx + d} = \frac{p}{q} \implies q(ax + b) = p(cx + d)

Clearing Denominators: Multiply by LCM of denominators\text{Multiply by } \text{LCM of denominators}

💡Examples

Problem 1:

Solve for xx: x2−15=x3+14\frac{x}{2} - \frac{1}{5} = \frac{x}{3} + \frac{1}{4}

Solution:

Step 1: Find the LCM of denominators 2, 5, 3, and 4, which is 60.\text{Step 1: Find the LCM of denominators 2, 5, 3, and 4, which is 60.} Step 2: Multiply every term by 60:\text{Step 2: Multiply every term by 60:} 60(x2)−60(15)=60(x3)+60(14)60(\frac{x}{2}) - 60(\frac{1}{5}) = 60(\frac{x}{3}) + 60(\frac{1}{4}) 30x−12=20x+1530x - 12 = 20x + 15 Step 3: Transpose 20x to LHS and -12 to RHS:\text{Step 3: Transpose } 20x \text{ to LHS and -12 to RHS:} 30x−20x=15+1230x - 20x = 15 + 12 10x=2710x = 27 Step 4: Divide by 10:\text{Step 4: Divide by 10:} x=2710 or 2.7x = \frac{27}{10} \text{ or } 2.7

Explanation:

To solve an equation with multiple fractions, we clear the denominators by multiplying the entire equation by the LCM of the denominators. This simplifies the equation into a standard linear form which can be solved by grouping like terms.

Problem 2:

Solve for xx: x+12x+3=38\frac{x + 1}{2x + 3} = \frac{3}{8}

Solution:

Step 1: Use cross-multiplication:\text{Step 1: Use cross-multiplication:} 8(x+1)=3(2x+3)8(x + 1) = 3(2x + 3) Step 2: Expand the brackets using the distributive law:\text{Step 2: Expand the brackets using the distributive law:} 8x+8=6x+98x + 8 = 6x + 9 Step 3: Transpose 6x to LHS and 8 to RHS:\text{Step 3: Transpose } 6x \text{ to LHS and 8 to RHS:} 8x−6x=9−88x - 6x = 9 - 8 2x=12x = 1 Step 4: Solve for x:\text{Step 4: Solve for } x: x=12x = \frac{1}{2}

Explanation:

When an equation consists of a single fraction on both sides, cross-multiplication is the most efficient method. We multiply the numerator of the LHS by the denominator of the RHS and vice versa, then solve the resulting linear equation.