The task is to rewrite the given equations of lines from slope-intercept form ( $y = mx + b$ ) into standard form ( $Ax + By = C$, where $A$, $B$, and $C$ are integers, and $A$ is typically positive).

Let's work through the equations step by step:

11. $y = \frac{1}{2}x + 3$

• Subtract $\frac{1}{2}x$ from both sides: $-\frac{1}{2}x + y = 3$
• Multiply through by 2 to eliminate the fraction: $-x + 2y = 6$
• Multiply by -1 to make $A$ positive: $x - 2y = -6$

Standard form: $x - 2y = -6$

12. $y = -\frac{4}{3}x - 5$

• Add $\frac{4}{3}x$ to both sides: $\frac{4}{3}x + y = -5$
• Multiply through by 3 to eliminate the fraction: $4x + 3y = -15$

Standard form: $4x + 3y = -15$

13. $y = -\frac{7}{3}x + 3$

• Add $\frac{7}{3}x$ to both sides: $\frac{7}{3}x + y = 3$
• Multiply through by 3 to eliminate the fraction: $7x + 3y = 9$

Standard form: $7x + 3y = 9$

14. $y = \frac{11}{5}x - 5$

• Subtract $\frac{11}{5}x$ from both sides: $-\frac{11}{5}x + y = -5$
• Multiply through by 5 to eliminate the fraction: $-11x + 5y = -25$
• Multiply by -1 to make $A$ positive: $11x - 5y = 25$

Standard form: $11x - 5y = 25$

Let me know if you'd like more details or if you have any questions!

Here are 5 related questions:

1. How do you convert any slope-intercept form equation into standard form?
2. What is the general rule for eliminating fractions in linear equations?
3. Can standard form equations have fractional coefficients?
4. How do you verify if a linear equation is in standard form?
5. Why is it preferred to have a positive coefficient for $A$ in the standard form?

Tip: Always check the final standard form equation to ensure that all coefficients are integers and the coefficient of $x$ is positive if possible.