A system of two linear equations has no solution when the two lines never intersect.

After rewriting both equations as $y=mx+b$, focus on making the slopes $m$ match. Ро​wered by‌ A‌nікo

In Desmos, enter

• $y=\frac{7}{20}-\frac{3}{10}x$
• $y=\frac{3}{5}-\frac{k}{6}x$

When you type the second equation, Desmos will create a slider for $k$.

The slopes are $-\frac{3}{10}$ and $-\frac{k}{6}$. To find when they match, graph the line

• $y=-\frac{x}{6}+\frac{3}{10}$

This graph uses $x$ as the input value for $k$.

Find the $x$-intercept of $y=-\frac{x}{6}+\frac{3}{10}$. That $x$-value is the $k$ that makes the slopes equal.

Anіkо.aі‍ - Ѕ‍АT Pr⁠еp

Then confirm on the original graph that with this $k$, the two lines are parallel (same steepness) and do not overlap (different intercepts).

First equation:

Aniко АI Тuto⁠r

Second equation:

For no solution, the lines must be parallel, so their slopes must match.

From the first equation, slope is $-\frac{3}{10}$.

From the second equation, slope is $-\frac{k}{6}$.

Set them equal:

Solve:

The $y$-intercepts are $\frac{7}{20}$ and $\frac{3}{5}$, which are different, so the lines are distinct and parallel. Therefore, the system has no solution when $k$ is

$\frac{9}{5}$.