To change a linear equation from point-slope form to slope-intercept form, you can follow some simple steps. Let's break it down so it's easy to understand.

What Are the Two Forms?

The point-slope form looks like this:

$$y - y_1 = m(x - x_1)$$

Here, $(x_1, y_1)$ is a point on the line, and $m$ is how steep the line is, called the slope.

The slope-intercept form is written like this:

$$y = mx + b$$

In this form, $m$ is still the slope, and $b$ shows where the line crosses the y-axis.

Steps to Change Forms

1. Start with the Point-Slope Form: Begin with an equation in point-slope form, like this one:

   $$y - 3 = 2(x - 1)$$

2. Distribute the Slope: Next, spread the slope $m$ (which is 2) into the equation. For our example, it looks like this:

   $$y - 3 = 2x - 2$$

3. Isolate the $y$ Term: To get it into slope-intercept form, we want $y$ by itself. So, add 3 to both sides:

   $$y = 2x - 2 + 3$$

4. Combine Like Terms: Now, we need to put together the similar numbers on the right side. That gives us:

   $$y = 2x + 1$$

5. Final Result: Now, we have $y = 2x + 1$, which is in slope-intercept form. Here, the slope $m$ is 2, and the y-intercept $b$ is 1.

Checking Your Work:

It’s a good idea to double-check each step to make sure everything is right:

• From $y - 3 = 2(x - 1)$: Confirm the slope $m=2$ and the point $(1, 3)$ are correct.
• Distributing gives $y - 3 = 2x - 2$: This step checks out.
• Adding 3 leads to $y = 2x + 1$: Here, $b=1$ shows where the line crosses the y-axis.

Why Is This Important?

Knowing how to change equations is helpful in real life and school:

• Graphing Lines: Understanding how slope and intercept work together helps us know how a line will look on a graph.
• Solving Problems: Linear equations are used in fields like economics, physics, and social studies. Changing forms can help understand different situations.

Common Mistakes to Avoid:

When switching from point-slope to slope-intercept form, watch out for these common errors:

• Wrong Signs: Make sure you check your signs when you distribute the slope.
• The Point: Always remember to include the point $(x_1, y_1)$ correctly.

More Examples:

1. Example 2:

   • Start with $y - 4 = -3(x + 2)$.
   • Distribute: $y - 4 = -3x - 6$.
   • Isolate $y$: $y = -3x - 6 + 4$.
   • Combine: $y = -3x - 2$.

2. Example 3:

   • Start with $y + 5 = \frac{1}{2}(x - 4)$.
   • Distribute: $y + 5 = \frac{1}{2}x - 2$.
   • Isolate $y$: $y = \frac{1}{2}x - 2 - 5$.
   • Combine: $y = \frac{1}{2}x - 7$.

Conclusion:

Changing equations between these forms is a key skill in algebra. It helps you understand relationships shown by linear functions. Whether for drawing graphs, solving real-world problems, or getting ready for more complex math, being able to change between point-slope and slope-intercept forms is really important.

Getting good at these steps isn’t just about memorizing them; it’s about understanding why you do them, which will help you think better and solve problems easier later on.