Questions and Answers
What is the solution for the equation $3(x - 3) = 3x - 6$?
no solution
What is the value of b in the equation $\frac{b}{3} - 1 = 5$?
b = 18
What is the value of k in the equation $4k - 5k = 10 + k$?
k = -5
What is the value of x in the equation $6(2x + 10) = 10(x + 6)$?
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What are the solutions for the equation $-2(x - 5) = 10 - 2x$?
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What is the solution for x in the equation $\frac{1}{5}x - \frac{2}{3} = \frac{1}{4}x + 1$?
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If Phil, Claire, and Alex combined spend $275, how much did Phil spend if Claire spends $25 more than Phil and Alex spends half as much as Phil?
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What are the dimensions of a rectangle with a perimeter of 82 cm, where the length is one more than three times the width?
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What are three consecutive odd integers whose sum is -39?
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How old are Rob, Charles, and Kristi if the sum of their ages is 105 years and Kristi is 5 years older than Charles, while Rob is 20 years younger than twice Charles' age?
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Study Notes
Equations and Solutions
- No Solution: When solving 3(x - 3) = 3x - 6, the result is that there is no solution.
- Value Identification: For the equation b/3 - 1 = 5, solving gives b = 18.
- Negative Result: The equation 4k - 5k = 10 + k results in k = -5.
- Zero Solution: In the equation 6(2x + 10) = 10(x + 6), the solution yields x = 0.
- Infinite Solutions: The equation -2(x - 5) = 10 - 2x has infinite solutions, indicating all real numbers are a solution.
- Fractional Solution: For 1/5 x - 2/3 = 1/4x + 1, the resulting value of x is -100/3 or approximately -33.33.
Word Problems and Equations
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Shopping Scenario: Phil, Claire, and Alex combined spend $275. Formulating an equation from their spending habits:
- Phil = x
- Claire = x + 25
- Alex = 1/2x
- Equation: x + (x + 25) + (1/2x) = 275 leads to Phil spending $100, Claire $125, and Alex $50.
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Rectangle Perimeter: For a rectangle with a perimeter of 82 cm, with length being one more than three times the width:
- Length = 3w + 1, Width = w
- Set up the equation 2(3w + 1) + 2w = 82 leads to dimensions of width = 10 cm and length = 31 cm.
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Odd Integers Sum: The sum of three consecutive odd integers equaling -39 can be structured:
- 1st integer: x, 2nd: x + 2, 3rd: x + 4
- Equation: x + (x + 2) + (x + 4) = -39 solves to integers -15, -13, and -11.
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Siblings' Ages: For siblings Rob, Charles, and Kristi with a combined age of 105:
- Charles = x, Kristi = x + 5, Rob = 2x - 20
- Using the equation 4x - 15 = 105 results in ages: Charles = 30 years, Kristi = 35 years, Rob = 40 years.
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Test your knowledge with these flashcards that cover key equations and word problems from Algebra 1 Unit 1. Each card presents a different problem to solve, helping you understand various solution types, including specific values and identities.
