Key Concepts and Formulas

• Intercept Form of a Line: The equation of a line with x-intercept $a$ and y-intercept $b$ is given by $\frac{x}{a} + \frac{y}{b} = 1$.
• Point on a Line: If a point $(x_1, y_1)$ lies on a line, then the coordinates of the point must satisfy the equation of the line.
• Solving Quadratic Equations: Remember to consider both positive and negative roots when solving equations of the form $x^2 = c$.

Step-by-Step Solution

• Step 1: Express the given condition on the intercepts. We are given that the sum of the x-intercept $a$ and the y-intercept $b$ is $-1$. Therefore, we have: $a + b = -1$ We can express $b$ in terms of $a$: $b = -1 - a \quad \text{(Equation 1)}$ Why this step? Expressing one variable in terms of the other allows us to substitute into the other equation and solve for a single variable.

• Step 2: Use the point (4, 3) to form another equation. Since the line passes through the point $(4, 3)$, we can substitute $x = 4$ and $y = 3$ into the intercept form of the equation: $\frac{x}{a} + \frac{y}{b} = 1$ $\frac{4}{a} + \frac{3}{b} = 1 \quad \text{(Equation 2)}$ Why this step? Any point on the line must satisfy the equation of the line. This gives us a second equation relating $a$ and $b$.

• Step 3: Substitute Equation 1 into Equation 2. Substitute $b = -1 - a$ into Equation 2: $\frac{4}{a} + \frac{3}{-1 - a} = 1$ Why this step? Substitution allows us to eliminate one of the variables and obtain an equation in terms of a single variable.

• Step 4: Simplify and solve for a. Multiply both sides by $a(-1-a)$ to clear the fractions: $4(-1 - a) + 3a = a(-1 - a)$ Expand and simplify: $-4 - 4a + 3a = -a - a^2$ $-4 - a = -a - a^2$ Add $a$ to both sides: $-4 = -a^2$ Multiply by -1: $4 = a^2$ Taking the square root of both sides: $a = \pm 2$ Why this step? This is standard algebraic manipulation to solve for the unknown variable $a$. Remember both positive and negative roots.

• Step 5: Find the corresponding values of b. Use Equation 1 ($b = -1 - a$) to find the corresponding values of $b$ for each value of $a$.

  Case 1: $a = 2$ $b = -1 - 2 = -3$ So, one pair of intercepts is $(a, b) = (2, -3)$.

  Case 2: $a = -2$ $b = -1 - (-2) = -1 + 2 = 1$ So, the second pair of intercepts is $(a, b) = (-2, 1)$. Why this step? We need to find the value of $b$ that corresponds to each value of $a$. Each pair $(a,b)$ will define a unique line.

• Step 6: Form the equations of the lines. Substitute the pairs of values $(a, b)$ into the intercept form $\frac{x}{a} + \frac{y}{b} = 1$.

  Line 1 (using $a = 2, b = -3$): $\frac{x}{2} + \frac{y}{-3} = 1$ $\frac{x}{2} - \frac{y}{3} = 1$

  Line 2 (using $a = -2, b = 1$): $\frac{x}{-2} + \frac{y}{1} = 1$ $-\frac{x}{2} + y = 1$ Why this step? This step translates the intercept values into the equations of the lines.

Common Mistakes & Tips

• Forgetting the Negative Root: When solving $a^2 = 4$, remember that $a$ can be both $+2$ and $-2$.
• Sign Errors: Be careful with signs when substituting and simplifying equations.
• Choosing the Correct Form: Using the intercept form is crucial for this problem.

Summary

We used the intercept form of a line and the given conditions to form a system of equations. Solving this system yielded two possible values for the x-intercept, and consequently, two equations for the lines that satisfy the given conditions. The equations are $\frac{x}{2} - \frac{y}{3} = 1$ and $\frac{x}{-2} + \frac{y}{1} = 1$, which corresponds to option (A).

Final Answer The final answer is \boxed{\frac{x}{2} - \frac{y}{3} = 1 \text{ and } \frac{x}{-2} + \frac{y}{1} = 1}, which corresponds to option (A).