Try rewriting each equation in the form $y = mx + b$ so you can compare their slopes and y-intercepts. Focus on isolating $y$ in each equation.

Once you have both slopes in terms of $k$, set the slopes equal to find the value of $k$ that makes the lines parallel. Then check that the y-intercepts are not equal so the lines are not the same line.

In Desmos, type the first equation exactly as given: 0.5x - (3/4)y = 2. This will draw the first line.

Type the second equation as k x - (9/2)y = 10. Desmos will prompt you to add a slider for k; accept it so you can change the value of $k$ dynamically.

Move the $k$ slider and watch how the second line moves. You are looking for the value of $k$ where the two lines are parallel (never intersect) but have different y-intercepts. When you see that the lines never cross and are distinct, read that value of $k$ from the slider.

For a system of two linear equations in $x$ and $y$, there is no solution when the two lines are parallel but not the same line. In algebra terms, that means the lines have the same slope but different y-intercepts.

Start with the first equation:

Solve for $y$:

• Move the constant: $\frac12 x - 2 = \frac34 y$
• Multiply both sides by $4$ to clear fractions:

• Solve for $y$:

So the slope of the first line is $m_1 = \frac{2}{3}$.

Now take the second equation:

Solve for $y$:

• Move the constant: $kx - 10 = \frac92 y$
• Multiply both sides by $2$ to clear the fraction:

• Solve for $y$:

So the slope of the second line is $m_2 = \frac{2k}{9}$, and its y-intercept is $-\frac{20}{9}$.

For the lines to be parallel (same slope), set the slopes equal:

Cross-multiply:

With this $k$, the first line has y-intercept $-\frac{8}{3}$ and the second line has y-intercept $-\frac{20}{9}$, and

So the lines are parallel and different, which means the system has no solution when $k = 3$.