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How do mathematicians solve equations where the mystery number appears on both sides of the equal sign? Linear equations in one variable are the supre...
How do mathematicians solve equations where the mystery number appears on both sides of the equal sign? Linear equations in one variable are the supreme problem-solving tools of algebra.
Why This Chapter Matters
In Class 8 Mathematics, "Algebra Play" delivers rigorous mathematical reasoning, algebraic precision, and geometric mastery aligned with the 2026–27 NCERT curriculum.
Before You Begin (Prerequisites)
- Linear equations from Class 7 ($ax + b = c$).
- Transposition rules (+ becomes -, $\times$ becomes $\div$).
- Removing brackets using distributivity.
What You Will Learn (Core Objectives)
- Solve linear equations with variables on both sides ($ax + b = cx + d$).
- Simplify equations with fractional coefficients and brackets using cross-multiplication.
- Translate complex word problems (ages, digits, geometry, coins) into linear equations.
- Check and verify solutions systematically.
- Apply linear modeling to real-world financial balance problems.
Chapter Roadmap & Progression
1
1. Equations with Variables on Both...
2
2. Equations Reducible to Linear Fo...
3
3. Word Problem Modeling Strategies
Complete Concept Guide (100% Curriculum Coverage)
1. Equations with Variables on Both Sides
To solve $5x + 9 = 2x + 24$, transpose all variable terms to the left side and all constants to the right: $5x - 2x = 24 - 9 \implies 3x = 15 \implies x = 5$. Check: $5(5) + 9 = 34$, and $2(5) + 24 = 34$. Verified!
2. Equations Reducible to Linear Form (Cross-Multiplication)
When an equation has fractions on both sides, such as $\frac{x + 1}{2x + 3} = \frac{3}{8}$, apply cross-multiplication: $8(x + 1) = 3(2x + 3) \implies 8x + 8 = 6x + 9 \implies 2x = 1 \implies x = \frac{1}{2}$.
3. Word Problem Modeling Strategies
Assign a single variable $x$ to the primary unknown. Express all other quantities in terms of $x$, set up the equality dictated by the story, and solve.
Check Your Understanding (Diagnostic Practice Questions)
Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.
1
Solve: $8x + 4 = 3(x - 1) + 7$
Reveal Answer & Explanation
Answer: $8x + 4 = 3x - 3 + 7 \implies 8x + 4 = 3x + 4 \implies 8x - 3x = 4 - 4 \implies 5x = 0 \implies x = 0$.
Expand bracket first.
2
Solve by cross-multiplication: $\frac{z}{z + 15} = \frac{4}{9}$
Reveal Answer & Explanation
Answer: $9z = 4(z + 15) \implies 9z = 4z + 60 \implies 5z = 60 \implies z = 12$.
Cross-multiply terms.
3
The ages of Hari and Harry are in the ratio $5 : 7$. Four years from now, their ages will be in the ratio $3 : 4$. Find their present ages.
Reveal Answer & Explanation
Answer: Let ages be $5x$ and $7x$. In 4 years: $\frac{5x + 4}{7x + 4} = \frac{3}{4} \implies 4(5x + 4) = 3(7x + 4) \implies 20x + 16 = 21x + 12 \implies x = 4$. Present ages: 20 years and 28 years.
Age ratio problem.
4
The sum of three consecutive multiples of 8 is 888. Find these multiples.
Reveal Answer & Explanation
Answer: Let multiples be $8x, 8x + 8, 8x + 16$. Sum: $24x + 24 = 888 \implies 24x = 864 \implies x = 36$. Multiples are 288, 296, 304.
Consecutive multiples of 8.
5
A positive number is 5 times another number. If 21 is added to both, one becomes twice the other. Find the numbers.
Reveal Answer & Explanation
Answer: Let numbers be $x$ and $5x$. $5x + 21 = 2(x + 21) \implies 5x + 21 = 2x + 42 \implies 3x = 21 \implies x = 7$. Numbers are 7 and 35.
Linear word problem.
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