(c) \( x+2 y=12 \)

It looks like you're referring to part (c) of a problem involving the linear equation: \[ x + 2y = 12 \] To assist you effectively, I'll provide a comprehensive analysis of this equation. If you have a specific question or context in which this equation is used, please let me know! ## 1. **Standard Form of a Linear Equation** The given equation is in **standard form**, which is generally written as: \[ Ax + By = C \] Where: - \( A, B, \) and \( C \) are constants. - \( x \) and \( y \) are variables. For your equation: - \( A = 1 \) - \( B = 2 \) - \( C = 12 \) ## 2. **Solving for One Variable** You can solve for either \( x \) or \( y \) to express the equation in **slope-intercept form** (\( y = mx + b \)), which is useful for graphing. ### a. **Solving for \( y \):** \[ x + 2y = 12 \] Subtract \( x \) from both sides: \[ 2y = -x + 12 \] Divide both sides by 2: \[ y = -\frac{1}{2}x + 6 \] So, the slope-intercept form is: \[ y = -\frac{1}{2}x + 6 \] **Interpretation:** - **Slope (\( m \))**: \( -\frac{1}{2} \) - **Y-intercept (\( b \))**: \( 6 \) ### b. **Solving for \( x \):** Alternatively, solve for \( x \): \[ x = -2y + 12 \] ## 3. **Finding Intercepts** ### a. **X-intercept ($x, 0$):** Set \( y = 0 \): \[ x + 2(0) = 12 \] \[ x = 12 \] **X-intercept:** \( (12, 0) \) ### b. **Y-intercept ($0, y$):** Set \( x = 0 \): \[ 0 + 2y = 12 \] \[ y = 6 \] **Y-intercept:** \( (0, 6) \) ## 4. **Graphing the Equation** To graph the equation \( x + 2y = 12 \): 1. **Plot the Intercepts:** - Plot the x-intercept at \( (12, 0) \). - Plot the y-intercept at \( (0, 6) \). 2. **Draw the Line:** - Connect these two points with a straight line. - Extend the line in both directions. 3. **Slope Interpretation:** - The slope \( -\frac{1}{2} \) indicates that for every increase of 2 units in \( x \), \( y \) decreases by 1 unit. **Graph Example:**  *(Please note that the image link is illustrative. In an actual setting, the graph would be drawn to scale with the intercepts marked.)* ## 5. **Solution Set** The equation \( x + 2y = 12 \) represents all the points \( (x, y) \) that lie on the line. Some example solutions include: - \( (12, 0) \) - \( (10, 1) \) - \( (8, 2) \) - \( (6, 3) \) - \( (4, 4) \) - \( (2, 5) \) - \( (0, 6) \) ## 6. **Applications** Linear equations like \( x + 2y = 12 \) are fundamental in various fields such as: - **Economics:** Modeling supply and demand. - **Physics:** Describing relationships between variables like distance, time, and speed. - **Engineering:** Designing systems with linear relationships. - **Everyday Problem Solving:** Budgeting, scheduling, etc. ## 7. **Additional Considerations** - **Parallel and Perpendicular Lines:** - **Parallel lines** have the same slope. Any line with a slope of \( -\frac{1}{2} \) is parallel to \( x + 2y = 12 \). - **Perpendicular lines** have slopes that are negative reciprocals. A line perpendicular to \( x + 2y = 12 \) would have a slope of \( 2 \). - **Systems of Equations:** - If this equation is part of a system with another linear equation, you can find the point of intersection by solving the system simultaneously. ## 8. **Example Problem Involving \( x + 2y = 12 \)** **Problem:** Find the point where the line \( x + 2y = 12 \) intersects with the line \( y = x - 2 \). **Solution:** 1. **Set the equations equal to each other:** \[ y = x - 2 \] \[ x + 2y = 12 \] 2. **Substitute \( y \) from the first equation into the second:** \[ x + 2(x - 2) = 12 \] \[ x + 2x - 4 = 12 \] \[ 3x - 4 = 12 \] \[ 3x = 16 \] \[ x = \frac{16}{3} \] 3. **Find \( y \):** \[ y = \frac{16}{3} - 2 = \frac{16}{3} - \frac{6}{3} = \frac{10}{3} \] **Intersection Point:** \( \left( \frac{16}{3}, \frac{10}{3} \right) \) ## 9. **Conclusion** The equation \( x + 2y = 12 \) is a linear equation representing a straight line in the Cartesian plane. Understanding its properties—such as slope, intercepts, and how to graph it—is essential for solving various mathematical problems. If you have a specific aspect of this equation you'd like to explore further or if it's part of a larger problem, feel free to provide more details!