Solve the following equation for x. -18x - 23 = -³⁷⁄₂

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When solving an equation with one variable, your goal is to isolate the variable on one side of the equal sign. You do this by moving all of the other numbers to the opposite of the equal sign. You move numbers by performing the opposite operation (subtraction/addition and multiplication/division), on both sides of the equation to keep it balanced. For this problem: The first thing you want to do is move the ( - 23) to the other side of the equation by adding 23 (opposite operation) to both sides. -18x - 23 + 23 = -³⁷⁄₂ + 23 -18x = -³⁷⁄₂ + 23 Now let's simplify the right side of the equation. Remember that any whole number in fractional form is that number over 1. -18x = -³⁷⁄₂ + 23 -18x = -³⁷⁄₂ + ²³⁄₁ Next, we need to add the fractions. In order to do this, we need to find a common denominator for both fractions. This will be the least common multiple between the two denominators. You can easily find this by listing the multiples of each number: 1: 1, 2, 3, 4, 5, 6… 2: 2, 4, 6, 8, 10, 12, 14, 16… The lowest common multiple between 1 and 2 is 2. In order for the fractions to have the same denominator, we need to multiply each fraction by a fractional version of "1" which is basically any number over itself (e.g. ⁵/₅= 1). In order to get from 1 to 2, we need to multiply it by 2. Therefore, we multiply the entire fraction by ²/₂. Since the other denominator is already 2, you do not need to multiply the fraction by any number. -18x = -³⁷⁄₂ + ²³⁄₁ -18x = -³⁷⁄₂ + (²³⁄₁ • ²⁄₂) -18x = -³⁷⁄₂ + ⁴⁶⁄₂ Now that you have a common denominator, you simply add the numerators. Remember that you are adding a negative number here, so it's actually 46-37. -18x = -³⁷⁄₂ + ⁴⁶⁄₂ -18x = ⁹⁄₂ Finally, you need to move the -18, which is multiplying x. You do this by dividing (opposite operation) both sides by -18. -18x ÷ -18 = ⁹⁄₂ ÷ -18 Any whole number written as a fraction is simply that number over 1. So we have: x = ⁹⁄₂ ÷ -¹⁸⁄₁ Simplify the right side of the equation. To divide fractions, we multiply by the reciprocal (flip the second fraction and multiply): x = ⁹⁄₂ ÷ -¹⁸⁄₁ x = ⁹⁄₂ • -¹⁄₁₈ Remember that you multiply across the fraction: multiply the two numerators and the two denominators. Since you are multiplying one negative number by one positive number, the answer will also be negative. x = -⁹⁄₃₆ Reduce the final fraction by finding the largest common factor between the numerator and denominator and divide both by that number. 9: 1, 3, 9 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 Divide both 9 and 36 by 9 to get your reduced fraction. x = -⁹⁄₃₆ x = -¹⁄₄

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