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

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!