IGP(TAGK AT HOWE) 1.) Given: \( 3 x+2 y=6 \) a.) Determine the \( x \) and \( y \)-intercapts b.) graph by atteast 5 IPaints table of value of \( x \) c.) Graph by slope intureept form \( y=m x t b \) d.) Graph by intercept form.

Sure, let's work through each part of the problem step by step. ### **Given Equation:** \[ 3x + 2y = 6 \] --- ### **a. Determine the \( x \)- and \( y \)-Intercepts** **1. \( x \)-Intercept:** - To find the \( x \)-intercept, set \( y = 0 \) and solve for \( x \). \[ 3x + 2(0) = 6 \\ 3x = 6 \\ x = 2 \] - **\( x \)-Intercept:** \( (2, 0) \) **2. \( y \)-Intercept:** - To find the \( y \)-intercept, set \( x = 0 \) and solve for \( y \). \[ 3(0) + 2y = 6 \\ 2y = 6 \\ y = 3 \] - **\( y \)-Intercept:** \( (0, 3) \) --- ### **b. Create a Table of Values Using at Least 5 Points** To graph the equation accurately, it's helpful to plot several points. We'll choose various \( x \)-values and solve for the corresponding \( y \)-values. **Given Equation:** \[ 3x + 2y = 6 \\ \Rightarrow 2y = 6 - 3x \\ \Rightarrow y = -\frac{3}{2}x + 3 \] **Table of Values:** | \( x \) | \( y \) Calculation | \( y \) Value | |---------|---------------------------------|---------------| | -2 | \( y = -\frac{3}{2}(-2) + 3 = 3 + 3 = 6 \) | 6 | | -1 | \( y = -\frac{3}{2}(-1) + 3 = 1.5 + 3 = 4.5 \) | 4.5 | | 0 | \( y = -\frac{3}{2}(0) + 3 = 0 + 3 = 3 \) | 3 | | 1 | \( y = -\frac{3}{2}(1) + 3 = -1.5 + 3 = 1.5 \) | 1.5 | | 2 | \( y = -\frac{3}{2}(2) + 3 = -3 + 3 = 0 \) | 0 | **Points to Plot:** - \((-2, 6)\) - \((-1, 4.5)\) - \((0, 3)\) - \((1, 1.5)\) - \((2, 0)\) --- ### **c. Graph Using Slope-Intercept Form \( y = mx + b \)** **Step 1: Convert to Slope-Intercept Form** \[ 3x + 2y = 6 \\ \Rightarrow 2y = -3x + 6 \\ \Rightarrow y = -\frac{3}{2}x + 3 \] - **Slope (\( m \))**: \( -\frac{3}{2} \) - **\( y \)-Intercept (\( b \))**: \( 3 \) **Step 2: Plot the Graph** 1. **Start at the \( y \)-intercept:** Plot the point \( (0, 3) \). 2. **Use the slope to find another point:** From \( (0, 3) \), move down 3 units and right 2 units to reach \( (2, 0) \). 3. **Draw the Line:** Connect the points with a straight line, extending in both directions. **Graph Representation:** ``` y | 6 | * 5 | 4 | * 3 | * 2 | 1 | * 0 |__________ x 0 1 2 ``` --- ### **d. Graph Using Intercept Form** **Intercept Form of a Line:** \[ \frac{x}{a} + \frac{y}{b} = 1 \] where \( a \) is the \( x \)-intercept and \( b \) is the \( y \)-intercept. **Given Equation:** \[ 3x + 2y = 6 \] **Convert to Intercept Form:** \[ \frac{3x}{6} + \frac{2y}{6} = 1 \\ \Rightarrow \frac{x}{2} + \frac{y}{3} = 1 \] - **\( x \)-Intercept (\( a \))**: \( 2 \) (from \( \frac{x}{2} \)) - **\( y \)-Intercept (\( b \))**: \( 3 \) (from \( \frac{y}{3} \)) **Step 1: Plot the Intercepts** - Plot \( (2, 0) \) and \( (0, 3) \). **Step 2: Draw the Line** - Connect the two intercepts with a straight line, extending in both directions. **Graph Representation:** ``` y | 6 | 5 | 4 | 3 | * 2 | 1 | 0 |*__________ x 0 1 2 ``` --- ### **Final Graph Combining All Methods** All three graphing methods should result in the same straight line passing through the points \( (2, 0) \) and \( (0, 3) \). Here's a simplified representation: ``` y | 6 | * 5 | 4 | * 3 | * 2 | 1 | * 0 |__________ x 0 1 2 ``` - **Key Points:** \((-2, 6)\), \((-1, 4.5)\), \((0, 3)\), \((1, 1.5)\), \((2, 0)\) - **Slope-Intercept Form:** \( y = -\frac{3}{2}x + 3 \) - **Intercept Form:** \( \frac{x}{2} + \frac{y}{3} = 1 \) All methods provide consistent ways to graph the equation \( 3x + 2y = 6 \).