If \(x = 4\), then \(y = -\frac{3}{4}(4) = -3\). If \(x = -4\), then \(y = -\frac{3}{4}(-4) = 3\).

["Exploring the Linear Relationship: Solving for (y) When (x = 4) and (x = -4)", "Mathematics often reveals elegant simplicity in how variables interact. One clear example involves solving for (y) in a simple linear equation: when (x = 4), (y = -\frac{3}{4}(4) = -3); and when (x = -4), (y = -\frac{3}{4}(-4) = 3). Let’s break down this straightforward calculation and explore its implications.", "The Equation: A Foundation of Proportionality", "The relationship expressed by ( y = -\frac{3}{4}x ) demonstrates a direct proportional dependency between (x) and (y), with a consistent negative scaling factor of (-\frac{3}{4}). This means:\n- When (x) increases, (y) decreases proportionally.\n- When (x) decreases, (y) increases proportionally.\n- The sign of (y) directly mirrors the sign of (x), adjusted by the negative coefficient.", "Calculating (y) for Specific (x) Values", "Consider the two cases presented:", "1. When (x = 4):\n Substituting into the equation:\n [\n y = -\frac{3}{4}(4) = -3\n ]\n This shows a straightforward application: multiplying (-3/4) by 4 yields (-3). The negative outcome reflects how the coefficient transforms positive input into a negative output.", "2. When (x = -4):\n Applying the same formula:\n [\n y = -\frac{3}{4}(-4) = 3\n ]\n Here, multiplying a negative (x) by the positive numerator yields a positive (y). The double negative cancels out, producing a positive result.", "Why This Relationship Matters", "Understanding such proportional relationships is especially useful in algebra, physics, engineering, and finance, where linear models describe trends and transformations. The consistent scaling factor (-\frac{3}{4}) illustrates how changes in input consistently scale (y) by that ratio—a core concept in linear thinking.", "Key Takeaways", "- The equation (y = -\frac{3}{4}x) represents a linear function with a slope of (-\frac{3}{4}).\n- Substituting specific (x) values demonstrates how linearity directly computes (y).\n- The sign rule (( \ ext{sign of } y = -(\ ext{sign of } x) )) is reliable with negative coefficients.\n- Such calculations reinforce logical reasoning and precision in mathematical problem-solving.", "Conclusion", "The simple expressions ( y = -\frac{3}{4}(4) = -3 ) and ( y = -\frac{3}{4}(-4) = 3 ) exemplify the clarity and predictability of linear equations. By grasping how input values transform through fixed coefficients, learners deepen their ability to model and interpret real-world scenarios with mathematical precision. Whether in basic algebra or advanced fields, mastering these foundational patterns remains essential.", "---", "Keywords: linear equations, proportionality, (y = -\frac{3}{4}x), solving for (y), algebraic calculations, mathematical relationships"]