Directions: Write the linear equation in Standard Form. \( y=-\frac{1}{3} x+\frac{23}{9} \) Incorrect Answer Given: \( -\frac{1}{3} x-y=-\frac{23}{9} \) Why is this incorrect? Answer fails to be in the form: Answer fails to make sure \( A, B, C \) are not fractions (or decimals). Answer fails to make sure \( A \) is positive.

To convert the equation \( y = -\frac{1}{3} x + \frac{23}{9} \) into standard form, we need to rearrange it into the form \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) is positive. 1. Start with the given equation: \[ y = -\frac{1}{3} x + \frac{23}{9} \] 2. To eliminate the fraction, we can multiply the entire equation by 9 (the least common multiple of the denominators): \[ 9y = 9 \left(-\frac{1}{3} x\right) + 9 \left(\frac{23}{9}\right) \] This simplifies to: \[ 9y = -3x + 23 \] 3. Rearranging this to standard form \( Ax + By = C \): \[ 3x + 9y = 23 \] 4. In this equation, \( A = 3 \), \( B = 9 \), and \( C = 23 \). All coefficients are integers, and \( A \) is positive. Thus, the correct standard form of the equation is: \[ 3x + 9y = 23 \] ### Why the Incorrect Answer is Incorrect: - The incorrect answer \( -\frac{1}{3} x - y = -\frac{23}{9} \) does not satisfy the standard form requirements: - It contains fractions, which is not allowed in standard form. - The coefficient \( A \) (which is \(-\frac{1}{3}\)) is not positive. Therefore, the correct standard form is \( 3x + 9y = 23 \).