Certainly! Let's analyze the linear equation you've provided: \[ 4x + y = 22 \] ### (a) Solving for $$ y $$ in Terms of $$ x $$ To better understand the equation, we can solve for $$ y $$ in terms of $$ x $$. This form is often useful for graphing the equation or understanding the relationship between the variables. \[ \begin{align*} 4x + y &= 22 \\ y &= 22 - 4x \end{align*} \] So, the equation can be rewritten as: \[ y = -4x + 22 \] ### Interpretation 1. **Slope-Intercept Form:** The equation $$ y = -4x + 22 $$ is in the slope-intercept form $$ y = mx + b $$, where: - **Slope ($$ m $$)**: $$ -4 $$ This indicates that for every unit increase in $$ x $$, $$ y $$ decreases by 4 units. - **Y-intercept ($$ b $$)**: $$ 22 $$ This is the point where the line crosses the y-axis (i.e., when $$ x = 0 $$, $$ y = 22 $$). 2. **Graphing the Equation:** If you were to graph this equation: - Start at the y-intercept $$ (0, 22) $$. - From there, use the slope to find another point. Since the slope is $$ -4 $$, you can go down 4 units and right 1 unit to reach $$ (1, 18) $$. - Draw a straight line through these points. 3. **Solution Set:** This equation represents a straight line with infinitely many solutions. Each solution is a pair $$ (x, y) $$ that satisfies the equation. For example: - If $$ x = 2 $$, then $$ y = 22 - 4(2) = 14 $$, so $$ (2, 14) $$ is a solution. - If $$ x = -3 $$, then $$ y = 22 - 4(-3) = 34 $$, so $$ (-3, 34) $$ is a solution. ### Summary - **Equation in Slope-Intercept Form:** $$ y = -4x + 22 $$ - **Slope:** $$ -4 $$ - **Y-intercept:** $$ 22 $$ - **Graph:** A straight line decreasing from left to right, crossing the y-axis at $$ (0, 22) $$. If you have more parts to this problem or need further assistance with different aspects, feel free to provide additional details! $$ y = -4x + 22 $$

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(a) $ 4 x+y=22 $

Ask by Schmidt Norris. in the United States

Jan 24,2025

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$ y = -4x + 22 $

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Let's dive into some fun facts about linear equations! Did you know that the concept of linear equations dates back to ancient civilizations? Babylonians were among the first to solve linear equations around 2000 BC! They used geometric concepts and had remarkable skills in managing these problems that laid the groundwork for modern algebra. Now, onto practical applications! Linear equations are super useful in everyday life. For example, if you're planning a party and need to buy food and drinks, you can set up a linear equation to determine how much of each item you can afford based on your budget. By knowing the cost per item, you can calculate the perfect amount to buy while staying within your limits!

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(a) \( 4 x+y=22 \)

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