(a) Find the intercepts of the graph of the following equation. (b) Graph the equation. \( 3 x+4 y=6 \) A. The \( x \)-intercept is (a) What is the \( x \)-intercept? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. B. There is no \( x \)-intercept.

To find the intercepts of the equation \( 3x + 4y = 6 \), we will determine both the \( x \)-intercept and the \( y \)-intercept. ### (a) Finding the intercepts **1. Finding the \( x \)-intercept:** The \( x \)-intercept occurs when \( y = 0 \). We can find it by substituting \( y = 0 \) into the equation: \[ 3x + 4(0) = 6 \] \[ 3x = 6 \] \[ x = 2 \] So, the \( x \)-intercept is \( (2, 0) \). **2. Finding the \( y \)-intercept:** The \( y \)-intercept occurs when \( x = 0 \). We can find it by substituting \( x = 0 \) into the equation: \[ 3(0) + 4y = 6 \] \[ 4y = 6 \] \[ y = \frac{6}{4} = \frac{3}{2} \] So, the \( y \)-intercept is \( (0, \frac{3}{2}) \). ### Summary of intercepts: - The \( x \)-intercept is \( (2, 0) \). - The \( y \)-intercept is \( (0, \frac{3}{2}) \). ### (b) Graphing the equation To graph the equation \( 3x + 4y = 6 \), we can plot the intercepts we found: 1. Plot the \( x \)-intercept \( (2, 0) \). 2. Plot the \( y \)-intercept \( (0, \frac{3}{2}) \). Next, draw a straight line through these two points. The line represents all the solutions to the equation \( 3x + 4y = 6 \). ### Final Answers: A. The \( x \)-intercept is \( (2, 0) \).