Let's solve for the slope and the y-intercept of the equation $5x - 3y = 15$ and then discuss how to sketch the graph.

Step 1: Put the equation in slope-intercept form (y = mx + b)

We want to express the equation in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Starting with the given equation: $5x - 3y = 15$

Solve for $y$:

1. Subtract $5x$ from both sides: $-3y = -5x + 15$
2. Divide every term by $-3$: $y = \frac{5}{3}x - 5$

Step 2: Identify the slope and the y-intercept

• The slope ($m$) is $\frac{5}{3}$.
• The y-intercept ($b$) is $-5$.

Step 3: Graphing the function

To graph the equation $y = \frac{5}{3}x - 5$:

1. Y-intercept: Start by plotting the point where the line crosses the y-axis, which is at $(0, -5)$.
2. Slope: The slope is $\frac{5}{3}$, meaning for every 3 units you move to the right (along the x-axis), you move up 5 units (along the y-axis). Starting from the y-intercept $(0, -5)$, move right 3 units to $(3, -5)$, then move up 5 units to $(3, 0)$, and so on.

By connecting these points with a straight line, you get the graph.

Summary:

• Slope: $\frac{5}{3}$
• Y-intercept: $-5$
• Graph: A line crossing the y-axis at $(0, -5)$ and rising with a slope of $\frac{5}{3}$.

Would you like me to sketch the graph for you? Feel free to ask for further clarification or details!

Additional Questions:

1. How do you find the x-intercept of this line?
2. How can you convert an equation to slope-intercept form if it's in a different format (like standard form)?
3. What is the meaning of slope in real-world problems?
4. How would the graph change if the slope was negative?
5. How can we determine the equation of a line if we know two points on it?

Tip: When you’re graphing a line, always start with the y-intercept, then use the slope to find another point.