- 105 = (n - 1) \times 7

["Understanding the Equation: 105 = (n – 1) × 7 – Solving & Applying the Algebra", "When you encounter the equation 105 = (n – 1) × 7, it might initially look like a simple algebraic expression, but it opens the door to solving a meaningful mathematical problem. Whether you’re a student learning algebra or a teacher explaining step-by-step reasoning, understanding how to solve this equation helps build foundational problem-solving skills.", "### What Does the Equation Mean?", "The equation 105 = (n – 1) × 7 defines a relationship between variables. Here, n is the unknown variable we want to find. Rearranging the equation reveals that 105 is equal to seven times one less than n. In other words, (n – 1) represents a reduced count of n, and multiplying this by 7 gives exactly 105.", "### Step-by-Step Solution", "Let’s solve the equation manually:", "1. Start with the original equation:\n $$\n 105 = (n – 1) \ imes 7\n $$", "2. Divide both sides by 7 to isolate the term:\n $$\n \frac{105}{7} = n - 1\n \Rightarrow 15 = n - 1\n $$", "3. Add 1 to both sides:\n $$\n 15 + 1 = n\n \Rightarrow n = 16\n $$", "### Why Does n Equal 16?", "Plugging n = 16 back into the original equation confirms the solution:\n$$\n(n - 1) \ imes 7 = (16 - 1) \ imes 7 = 15 \ imes 7 = 105\n$$", "This verifies that the equation holds true when n = 16.", "### Real-World Applications & Teaching Value", "This type of equation models many real-world situations, such as:", "- Grouping or partitioning problems, like dividing resources where one item is always "missing" or reduced by one.\n- Pattern recognition, where values follow a sequence based on a base number multiplied by 7.\n- Educational exercises, teaching students how to isolate variables, apply inverse operations, and verify solutions.", "### Practice Problems to Reinforce Learning", "To solidify your understanding, try solving similar equations:\n- 84 = (x – 2) × 7\n- 112 = (y + 2) × 8\n- 63 = (z – 4) × 3", "These exercises help build fluency in manipulating algebraic expressions and encourage logical thinking.", "### Summary", "The equation 105 = (n – 1) × 7 is a concise but powerful expression that demonstrates key algebraic principles: isolation of variables, inverse operations, and verification. Solving it step-by-step not only confirms n = 16, but also strengthens essential mathematical reasoning skills applicable in science, finance, engineering, and everyday problem-solving.", "Whether you’re memorizing the solution or uncovering the logic behind it — mastering equations like this is a big step forward in mathematical confidence and competence.", "---", "Keywords: algebra, solving equations, 105 = (n – 1) × 7, linear equations, solving for n, mathematical reasoning, middle school math, algebra tutorial, step-by-step solving, educational math problems"]