Which one of the following statements is false for the following pair of linear equations?
$a_1x + b_1y + c_1 = 0$ $a_2x + b_2y + c_2 = 0$

Q: Which one of the following statements is false for the following pair of linear equations? $a_1x + b_1y + c_1 = 0$ $a_2x + b_2y + c_2 = 0$

Option 3 is the correct answer because it states that if $\frac{a_1}{a_2} = \frac{b_1}{b_2}$, then the lines will be intersecting. This statement is false. When $\frac{a_1}{a_2} = \frac{b_1}{b_2}$, it means that the lines are either coincident or parallel, depending on the value of $\frac{c_1}{c_2}$. If $\frac{c_1}{c_2}$ is also equal to $\frac{a_1}{a_2}$ and $\frac{b_1}{b_2}$, the lines are coincident. If $\frac{c_1}{c_2}$ is not equal, the lines are parallel. Therefore, the statement in Option 3 is false because it incorrectly claims that the lines will be intersecting. Now, let's explain why each of the other options is incorrect: 1. **Option 1**: This statement is true. If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the two lines are coincident, meaning they lie on top of each other and have infinitely many points in common. 2. **Option 2**: This statement is true. If $\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$, the lines have different slopes and will intersect at exactly one point. 4. **Option 4**: This statement is true. If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}$, the lines are parallel. They have the same slope but different y-intercepts, so they will never meet. In summary, Option 3 is the only false statement because it incorrectly describes the condition under which the lines would intersect. The other options correctly describe the relationships between the coefficients and the nature of the lines.

Question

Accepted Answer

Option 3 is the correct answer because it states that if $\frac{a_1}{a_2} = \frac{b_1}{b_2}$, then the lines will be intersecting. This statement is false. When $\frac{a_1}{a_2} = \frac{b_1}{b_2}$, it means that the lines are either coincident or parallel, depending on the value of $\frac{c_1}{c_2}$. If $\frac{c_1}{c_2}$ is also equal to $\frac{a_1}{a_2}$ and $\frac{b_1}{b_2}$, the lines are coincident. If $\frac{c_1}{c_2}$ is not equal, the lines are parallel. Therefore, the statement in Option 3 is false because it incorrectly claims that the lines will be intersecting.

Now, let's explain why each of the other options is incorrect:

1. **Option 1**: This statement is true. If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the two lines are coincident, meaning they lie on top of each other and have infinitely many points in common.

2. **Option 2**: This statement is true. If $\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$, the lines have different slopes and will intersect at exactly one point.

4. **Option 4**: This statement is true. If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}$, the lines are parallel. They have the same slope but different y-intercepts, so they will never meet.

In summary, Option 3 is the only false statement because it incorrectly describes the condition under which the lines would intersect. The other options correctly describe the relationships between the coefficients and the nature of the lines.