Let's go through each problem from the image step by step:

a. Solve: $4x - 5 = -5$

To solve for $x$, add 5 to both sides:

$4x - 5 + 5 = -5 + 5$

$4x = 0$

Now, divide both sides by 4:

$x = \frac{0}{4} = 0$

So, $x = 0$.

b. Find the $x$-intercept: $4x - 5y = -5$

The $x$-intercept occurs when $y = 0$. Substitute $y = 0$ into the equation:

$4x - 5(0) = -5$

$4x = -5$

Now, divide both sides by 4:

$x = \frac{-5}{4}$

So, the $x$-intercept is $x = -\frac{5}{4}$ or $-1.25$.

c. Find $y$ when $x = 0$: $4x - 5y = -5$

Substitute $x = 0$ into the equation:

$4(0) - 5y = -5$

$-5y = -5$

Now, divide both sides by $-5$:

$y = 1$

So, $y = 1$.

d. Solve for $y$: $4x - 5y = -5$

To solve for $y$, isolate $y$ on one side of the equation:

$4x - 5y = -5$

Subtract $4x$ from both sides:

$-5y = -4x - 5$

Now, divide both sides by $-5$:

$y = \frac{-4x - 5}{-5}$

Simplify:

$y = \frac{4}{5}x + 1$

So, $y = \frac{4}{5}x + 1$.

e. Graph: $4x - 5y = -5$

This is the equation of a straight line. To graph it, use the intercepts found earlier:

• The $x$-intercept is $x = -\frac{5}{4}$ or $x = -1.25$,
• The $y$-intercept is $y = 1$.

You can plot these two points on the graph:

• $(-1.25, 0)$
• $(0, 1)$

Then, draw a line through these two points.

Would you like any further details or help with another step? Here are some related questions:

1. How do we check if a solution is correct for a linear equation?
2. Can you explain how to calculate intercepts of more complex functions?
3. What is the significance of the slope in the equation $y = \frac{4}{5}x + 1$?
4. How would we solve this equation graphically using a different method?
5. What would happen if the equation had a higher degree (like a quadratic)?

Tip: When graphing a line, the slope is crucial. In the equation $y = \frac{4}{5}x + 1$, the slope $\frac{4}{5}$ tells us that for every 5 units we move horizontally (along the $x$-axis), we move 4 units vertically (along the $y$-axis).