Understanding Parallel Lines in Linear Equations

A pair of linear equations in two variables represents a pair of lines in a coordinate plane. These lines can be intersecting, parallel, or coincident.

For two linear equations given in the standard form:

\(a_1x + b_1y + c_1 = 0\)

\(a_2x + b_2y + c_2 = 0\)

The conditions for the lines to be parallel are when the ratio of the coefficients of \(x\) and the coefficients of \(y\) are equal, but this ratio is not equal to the ratio of the constant terms. Mathematically, this is expressed as:

\(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)

Analyzing the Given Linear Equations

We are given the following pair of linear equations:

1. \(4x + 6y - 1 = 0\)
2. \(2x + ky - 7 = 0\)

Comparing these equations with the standard form, we can identify the coefficients:

Applying the Parallel Lines Condition to Find k

For the lines to be parallel, the condition \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) must be satisfied.

Let's first use the equality part of the condition: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \)

Substitute the values from our equations:

\(\frac{4}{2} = \frac{6}{k}\)

Simplify the left side:

\(2 = \frac{6}{k}\)

Now, we solve for \(k\). Multiply both sides by \(k\) (assuming \(k \neq 0\)):

\(2k = 6\)

Divide both sides by 2:

\(k = \frac{6}{2}\)

\(k = 3\)

Verifying the Non-Equality Condition

Now we must check if, with \(k = 3\), the condition \( \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) holds true.

Using \(b_1 = 6\), \(b_2 = k = 3\), \(c_1 = -1\), and \(c_2 = -7\):

Calculate \( \frac{b_1}{b_2} \):

\(\frac{6}{3} = 2\)

Calculate \( \frac{c_1}{c_2} \):

\(\frac{-1}{-7} = \frac{1}{7}\)

Now, check if \( \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \):

\(2 \neq \frac{1}{7}\)

This is true. The ratio \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = 2 \), and this is not equal to \( \frac{c_1}{c_2} = \frac{1}{7} \). Thus, the condition for parallel lines is satisfied when \(k = 3\).

Conclusion

The value of \(k\) for which the given pair of linear equations represents parallel lines is \(k = 3\).

Revision Table: Conditions for Pairs of Linear Equations

Additional Information: Systems of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. When dealing with two linear equations in two variables, the solution to the system corresponds to the point(s) where the lines represented by the equations intersect.

• If the lines intersect at a single point, there is a unique solution. This happens when \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \).
• If the lines are coincident (one lies exactly on top of the other), they intersect at every point, resulting in infinitely many solutions. This occurs when \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \).
• If the lines are parallel, they never intersect, meaning there is no common point that satisfies both equations simultaneously. Hence, there is no solution to the system. This is the case when \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \).

The problem asks for the condition where the lines are parallel, which directly relates to the ratio of coefficients as discussed.