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:** ![Graph of x + 2y = 12](https://i.imgur.com/oBqYQwP.png) *(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! The equation $$ x + 2y = 12 $$ represents a straight line with a slope of $$ -\frac{1}{2} $$ and a y-intercept at $$ (0, 6) $$. Some example solutions include $$ (12, 0) $$ and $$ (10, 1) $$.

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Home Question Bank Math Algebra

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Ask by Bradley Henry. in the United States

Jan 24,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The equation

represents a straight line with a slope of

and a y-intercept at

. Some example solutions include

and

Solution

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Beyond the Answer

To solve the equation

, you can express

in terms of

. Rearranging gives

, leading to

. This equation represents a straight line when plotted on a graph, with

-intercept at (12, 0) and

-intercept at (0, 6). So, for every unit increase in

decreases by half a unit!

Let’s not forget about graphing this equation! It can be a fun activity to visualize how

and

interact. Just pick some values for

(like 0, 4, 8, 12) and solve for

each time. You’ll not only practice your algebra, but you’ll also get to draw a straight line, showcasing that awesome relationship between

and

in a tangible way!

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