- Combined like terms to simplify various algebraic expressions systematically.
- Isolated variables in linear equations using inverse operations to find precise solutions.
- Substituted given values \(k=2, m=5, n=3\) into expressions and followed the order of operations to compute numerical results.
- Translated word problems into algebraic equations and solved for the unknowns, finding:
- 5) \(\boxed{23}\)
- 6) \(\boxed{25}\)
- 7) \(\boxed{621}\)
- 8) \(\boxed{31}\)
Explanation
- Simplification and Algebraic Manipulation
We approach these problems systematically by addressing each category: algebraic simplification, equation solving, evaluation, and word problems.
- Simplification:
A\(4t + 3t = (4+3)t = 7t\)
B\(12y + 3y - 10y = (12+3-10)y = 5y\)
C\(5m - 3a + 4m - 2a = (5+4)m + (-3-2)a = 9m - 5a\)
D\(17z - 24z + 15z = (17-24+15)z = 8z\)
E\(15m - 4k + 5m - 6k = (15+5)m + (-4-6)k = 20m - 10k\)
f) \(40b + 7c - 12b = (40-12)b + 7c = 28b + 7c\)
- Solving Linear Equations
- Solving Linear Equations:
A\(5(\frac{x}{3} - 4) = 10 \implies \frac{x}{3} - 4 = 2 \implies \frac{x}{3} = 6 \implies x = 18\)
B\(4y - 18 = 2 \implies 4y = 20 \implies y = 5\)
C\(9t + 3t - 4 = 20 \implies 12t = 24 \implies t = 2\)
D\(3k + 2.5 = 10 \implies 3k = 7.5 \implies k = 2.5\)
E\(5y - 4 = 3y \implies 2y = 4 \implies y = 2\)
f) \(9b + 5.5 = 68.5 \implies 9b = 63 \implies b = 7\)
- Numerical Evaluation of Expressions
- Numerical Evaluation:
Given \(k = 2\), \(m = 5\), \(n = 3\):
A\(15k - m^2 + 2n = 15(2) - 5^2 + 2(3) = 30 - 25 + 6 = 11\)
B\(14n - mnk = 14(3) - (5)(3)(2) = 42 - 30 = 12\)
C\(mk + mn - 2nk = (5)(2) + (5)(3) - 2(3)(2) = 10 + 15 - 12 = 13\)
D\(\frac{4mnk + 80}{5} = \frac{4(5)(3)(2) + 80}{5} = \frac{120 + 80}{5} = 40\)
E\(\sqrt{4mnk + 24} = \sqrt{4(5)(3)(2) + 24} = \sqrt{144} = 12\)
f) \(3k + (2m - 3n) = 3(2) + (2(5) - 3(3)) = 6 + (10 - 9) = 7\)
g) \(4(k + m) - 2(m + n) = 4(2 + 5) - 2(5 + 3) = 28 - 16 = 12\)
h) \(4m - 3n - 2k = 4(5) - 3(3) - 2(2) = 20 - 9 - 4 = 7\)
i) \(\frac{1}{4}(4m - 2k) = \frac{1}{4}(4(5) - 2(2)) = \frac{16}{4} = 4\)
j) \(n^2 + m^2 - 5k^2 = 3^2 + 5^2 - 5(2^2) = 9 + 25 - 20 = 14\)
- Simplifying Algebraic Expressions
- Further Simplification:
A\(21d - 3(7 - 4d) = 21d - 21 + 12d = 33d - 21\)
B\(\frac{3}{4}(20y + 8w - 12z) = 15y + 6w - 9z\)
C\(\frac{32y^2st^3}{8y^2st} = \frac{32}{8} \cdot \frac{y^2}{y^2} \cdot \frac{s}{s} \cdot \frac{t^3}{t} = 4t^2\)
D\(\frac{8r^3t^4}{r^2t^3} = \frac{8}{1} \cdot \frac{r^3}{r^2} \cdot \frac{t^4}{t^3} = 8rt\)
- Word Problems and Summary
- Word Problems:
5) \(x + 17 = 40 \implies x = 23\)
6) \(x - 12 = 13 \implies x = 25\)
7) Total Score / 9 = 69 \(\implies\) Total = \(69 \times 9 = 621\)
8) \(2x + 15 = 77 \implies 2x = 62 \implies x = 31\)
Final answers summarized:
1) a: \(7t\), b: \(5y\), c: \(9m-5a\), d: \(8z\), e: \(20m-10k\), f: \(28b+7c\)
2) a: \(18\), b: \(5\), c: \(2\), d: \(2.5\), e: \(2\), f: \(7\)
3) a: \(11\), b: \(12\), c: \(13\), d: \(40\), e: \(12\), f: \(7\), g: \(12\), h: \(7\), i: \(4\), j: \(14\)
4) a: \(33d-21\), b: \(15y+6w-9z\), c: \(4t^2\), d: \(8rt\)
5) \(23\), 6) \(25\), 7) \(621\), 8) \(31\)
Examples
Algebraic simplification and solving linear equations are fundamental skills used in daily life for resource management and budgeting. For example, if you are planning a budget for a school project, you might have fixed costs (like material kits) and variable costs (like hourly labor). Representing these as algebraic expressions allows you to easily adjust your plan when the number of kits or hours changes, ensuring your total cost stays within your limits. Understanding how to solve these equations helps you find the exact amount of budget required for a target output, just as in the word problems provided.