Solve the equation. Express numbers as integers or simplified fractions.
$-\frac{3}{4} t-5=-\frac{9}{10} t+\frac{2}{5}$
The solution set is $\square$
Answer (2)
To solve the equation − 4 3 t − 5 = − 10 9 t + 5 2 , we start by grouping the terms involving t , then isolating t by using common denominators and simple arithmetic. The solution is t = 36 .
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Add 10 9 t to both sides and combine like terms: 20 3 t − 5 = 5 2 .
Add 5 to both sides: 20 3 t = 5 27 .
Multiply both sides by 3 20 : t = 3 20 ⋅ 5 27 .
Simplify to find the solution: t = 36 .
Explanation
Problem Analysis We are given the equation − 4 3 t − 5 = − 10 9 t + 5 2 and our goal is to solve for t .
Adding 10 9 t to both sides First, let's add 10 9 t to both sides of the equation to group the terms containing t together: − 4 3 t − 5 + 10 9 t = − 10 9 t + 5 2 + 10 9 t This simplifies to − 4 3 t + 10 9 t − 5 = 5 2
Combining terms with t Now, let's combine the terms with t . To do this, we need a common denominator for the fractions 4 3 and 10 9 . The least common multiple of 4 and 10 is 20, so we rewrite the fractions with a denominator of 20: − 4 3 = − 4 × 5 3 × 5 = − 20 15 10 9 = 10 × 2 9 × 2 = 20 18 So the equation becomes − 20 15 t + 20 18 t − 5 = 5 2 20 3 t − 5 = 5 2
Adding 5 to both sides Next, we add 5 to both sides of the equation: 20 3 t − 5 + 5 = 5 2 + 5 This simplifies to 20 3 t = 5 2 + 5
Combining constants Now, we need to combine the terms on the right side. To do this, we rewrite 5 as a fraction with a denominator of 5: 5 = 5 5 × 5 = 5 25 So the equation becomes 20 3 t = 5 2 + 5 25 20 3 t = 5 27
Isolating t To solve for t , we multiply both sides of the equation by 3 20 : 3 20 × 20 3 t = 3 20 × 5 27 This simplifies to t = 3 × 5 20 × 27 t = 15 540
Simplifying Finally, we simplify the fraction: t = 15 540 = 36 Thus, the solution is t = 36 .
Final Answer The solution set is t = 36 .
Examples
Imagine you're managing a small business and need to determine the break-even point for a new product. The equation we solved is similar to those used in cost-benefit analysis, where you equate costs and revenues to find the point where they balance. By solving such equations, you can find the exact number of units you need to sell to cover your expenses, helping you make informed decisions about pricing, production, and marketing strategies. This type of algebraic problem is a fundamental tool in business planning and financial analysis.
