Question

The x-intercept and y-intercept of a line are three times and four times of the x-intercept and y-intercept of the line $3x+2y=6$, respectively. Then the equation of the line is

• A. $2x-y=12$
• B. $2x+y=12$
• C. $2x-y=-12$
• D. $x+2y=12$
• E. $x+2y=12$

Hint:

Logic Tip: You can also find intercepts quickly by setting the other variable to 0. For $3x+2y=6$: set $y=0 \implies 3x=6 \implies x=2$. Set $x=0 \implies 2y=6 \implies y=3$. Then multiply by 3 and 4 respectively to get the new intercepts 6 and 12.

Answer 1

The Correct Option is B

Concept:
The intercept form of a line's equation is $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ is the x-intercept and $b$ is the y-intercept.
Step 1: Find the intercepts of the original line.
The given line is $3x + 2y = 6$. To convert this to the intercept form, divide the entire equation by 6: $$\frac{3x}{6} + \frac{2y}{6} = \frac{6}{6}$$ $$\frac{x}{2} + \frac{y}{3} = 1$$ From this, the x-intercept ($a$) is $2$, and the y-intercept ($b$) is $3$.
Step 2: Calculate the intercepts for the new line.
Let the intercepts of the new line be $A$ and $B$. The problem states: $A = 3 \cdot a = 3 \cdot 2 = 6$ $B = 4 \cdot b = 4 \cdot 3 = 12$
Step 3: Formulate the equation of the new line.
Using the intercept form $\frac{x}{A} + \frac{y}{B} = 1$: $$\frac{x}{6} + \frac{y}{12} = 1$$
Step 4: Convert to standard form.
Multiply the entire equation by the least common multiple (12) to clear the denominators: $$12 \cdot \left(\frac{x}{6}\right) + 12 \cdot \left(\frac{y}{12}\right) = 12 \cdot 1$$ $$2x + y = 12$$