Once you know the slope of the given line, remember that a perpendicular line has a slope that is the negative reciprocal. How do you find the negative reciprocal of a fraction?

You now know the slope of line $\ell$ and that it passes through $(2,-3)$. Use point-slope form $y - y_1 = m(x - x_1)$ to write its equation.

After you have the equation of line $\ell$, plug in $x = 0$ to find the $y$-coordinate of the point where it intersects the $y$-axis. That value is $b$.

Type 5x - 2y = 7 into Desmos. Then, separately type y = (5/2)x - 7/2 to see the same line in slope-intercept form and confirm that its slope is $5/2$.

Type the equation y + 3 = (-2/5)(x - 2) into Desmos. This uses the perpendicular slope $-2/5$ and the point $(2,-3)$ in point-slope form; Desmos will graph line $\ell$.

On the graph of line $\ell$, tap or click where the line crosses the $y$-axis. Desmos will display the coordinates of this point $(0, b)$; the $y$-value shown is the value of $b$.

Rewrite the equation $5x - 2y = 7$ in slope-intercept form $y = mx + b$.

Start by isolating $y$:

So the slope of the given line is $m = \frac{5}{2}$.

For two non-vertical lines to be perpendicular, their slopes must be negative reciprocals of each other. That means they multiply to $-1$.

The slope of the given line is $\frac{5}{2}$, so the perpendicular slope is obtained by:

• Flipping the fraction: $\frac{5}{2} \to \frac{2}{5}$
• Changing the sign: $\frac{2}{5} \to -\frac{2}{5}$

So the slope of line $\ell$ is $m_\ell = -\frac{2}{5}$.

Line $\ell$ passes through $(2,-3)$ and has slope $-\frac{2}{5}$. Use point-slope form:

Substitute $m = -\frac{2}{5}$ and $(x_1, y_1) = (2,-3)$:

This simplifies to:

Now isolate $y$ by subtracting $3$ from both sides:

Keep this expression; we will simplify the constant term next.

To find where line $\ell$ intersects the $y$-axis, set $x = 0$ in the equation from the previous step. First, simplify the constant term:

Write $3$ as $\frac{15}{5}$:

So the equation of line $\ell$ is:

At the $y$-axis, $x = 0$, so $y = -\frac{11}{5}$. Therefore, $b = -\frac{11}{5}$.