y - 2 = -\frac{3}{4} \left(x - \frac{3}{2} \right)

["# Solving the Linear Equation: A Step-by-Step Guide to y = -\frac{3}{4} \left(x - \frac{3}{2} \right)", "Understanding linear equations is fundamental in algebra, especially when working with functions and graphs. One essential form is the point-slope form of a linear equation, which is vital in modeling relationships between variables. In this article, we’ll explore the equation:", "[\ny = -\frac{3}{4} \left(x - \frac{3}{2} \right)\n]", "We’ll break down how to interpret and solve this linear equation, analyze its slope, and provide practical context for its use.", "---", "## What is the Point-Slope Form?", "The point-slope form of a linear equation expresses y in terms of (x) using a slope and a point on the line. The formula is:", "[\ny - y_1 = m(x - x_1)\n]", "where:\n- (m) is the slope of the line,\n- ((x_1, y_1)) is a known point on the line.", "This form is particularly useful when you know the slope and one point corresponding to the line.", "---", "## Rewriting the Given Equation", "The equation provided is:", "[\ny = -\frac{3}{4} \left(x - \frac{3}{2} \right)\n]", "Comparing this to the standard point-slope form:\n- The slope ((m)) is (-\frac{3}{4}),\n- The (x)-coordinate of a known point is (x_1 = \frac{3}{2}),\n- To find (y), plug in (x = \frac{3}{2}), which gives (y = 0). Hence, another key point is (\left(\frac{3}{2}, 0\right)).", "Thus, the equation describes a straight line passing through (\left(\frac{3}{2}, 0\right)) with slope (-\frac{3}{4}).", "---", "## Converting to Slope-Intercept Form", "While point-slope form is convenient, converting it to slope-intercept form ((y = mx + b)) often clarifies the graph’s properties like slope and y-intercept.", "Start with:", "[\ny = -\frac{3}{4} \left(x - \frac{3}{2} \right)\n]", "Distribute the (-\frac{3}{4}):", "[\ny = -\frac{3}{4}x + \frac{3}{4} \cdot \frac{3}{2}\n]", "Calculate:", "[\n\frac{3}{4} \cdot \frac{3}{2} = \frac{9}{8}\n]", "So,", "[\ny = -\frac{3}{4}x + \frac{9}{8}\n]", "Now, the equation is in standard form, showing:\n- Slope (m = -\frac{3}{4}): the line drops 3 units vertically for every 4 units forward horizontally.\n- Y-intercept (b = \frac{9}{8}): the line crosses the y-axis at (\left(0, \frac{9}{8}\right)).", "---", "## Graphing the Line", "To graph (y = -\frac{3}{4}x + \frac{9}{8}):", "1. Plot the y-intercept: Start at ((0, \frac{9}{8}) = (0, 1.125)).\n2. Use the slope: From the intercept, move down 3 units and right 4 units to reach ((4, -1.875)).\n3. Draw a straight line through these points.", "This visual representation helps interpret real-world scenarios like declining trends, slopes of gradients, or rates of change.", "---", "## Real-World Applications", "Linear equations like (y = -\frac{3}{4}x + \frac{9}{8}) appear in:", "- Economics: Modeling depreciation of assets or cost functions.\n- Physics: Describing velocity (negative slope) and motion at a constant rate.\n- Graphing Data: Analyzing trends in statistics, such as temperature changes over time.", "Using the point-slope form makes calculating and understanding these relationships intuitive and efficient.", "---", "## Solving for Specific Values of (y)", "To find (y) for any (x), use the original point-slope form:", "[\ny = -\frac{3}{4} \left(x - \frac{3}{2} \right)\n]", "For example:", "- When (x = 1):\n [\n y = -\frac{3}{4} \left(1 - \frac{3}{2} \right) = -\frac{3}{4} \cdot \left(-\frac{1}{2}\right) = \frac{3}{8}\n ]", "- When (x = \frac{3}{2}): (y = 0) (by construction).", "---", "## Summary", "- The equation (y = -\frac{3}{4} \left(x - \frac{3}{2} \right)) is a point-slope form linear equation.\n- It has a slope of (-\frac{3}{4}), indicating a downward trend.\n- Converting to slope-intercept form reveals the line’s y-intercept at (\left(0, \frac{9}{8}\right)).\n- Graphing helps visualize relationships and rates of change.\n- Real applications span science, economics, and data analysis.", "Mastering this equation equips you with a core tool for solving and graphing linear relationships in algebra and beyond.", "---", "## Further Resources", "- Practice converting between forms using algebra worksheets.\n- Use graphing calculators or apps to visualize changes in slope and intercept.\n- Explore real-world datasets and model them with linear equations.", "---\nUnlock algebraic clarity—start with mastering equations like this one!"]