91Ó°ÊÓ

The slope-intercept form of a linear equation is a very useful way to write an equation. It's written as:

\( y = mx + b \)

Here, 'm' represents the slope, and 'b' stands for the y-intercept. The slope 'm' indicates how steep the line is. For example, a slope of 3 means the line rises 3 units for every 1 unit it runs to the right. The y-intercept 'b' is the point where the line crosses the y-axis, which happens when x=0.

To convert a standard form equation to slope-intercept form, solve for y. Let's take the given equation:

\( \frac{2}{3} y - \frac{1}{2} x = 400 \)

First, move the x-term to the other side:
\( \frac{2}{3} y = \frac{1}{2} x + 400 \)

Next, multiply all terms by \( \frac{3}{2} \) to isolate y:
\( y = \frac{3}{4} x + 600 \)

Now the equation is in slope-intercept form. This helps us quickly identify the slope and y-intercept.

The x-intercept of a line is where the line crosses the x-axis. At this point, the y-coordinate is zero because it's on the x-axis. Finding the x-intercept involves setting y to zero in the equation and solving for x.

In our equation \( y = \frac{3}{4} x + 600 \), set y to 0:
\( 0 = \frac{3}{4} x + 600 \)

Now, solve for x:
Subtract 600 from both sides:
\( -600 = \frac{3}{4} x \)

Then, multiply both sides by \( \frac{4}{3} \):
\( x = -800 \)

So, the x-intercept is at the point (-800, 0). This tells us the line crosses the x-axis at x = -800.

The y-intercept of a line is where the line crosses the y-axis. At this point, the x-coordinate is zero because it's on the y-axis. Finding the y-intercept involves setting x to zero in the equation and solving for y.

Using our equation \( y = \frac{3}{4} x + 600 \), set x to 0:
\( y = \frac{3}{4} (0) + 600 = 600 \)

So, the y-intercept is at the point (0, 600). This tells us the line crosses the y-axis at y = 600. Identifying y-intercepts makes it easier to start plotting points on a graph.

When graphing a linear equation, it's super helpful to plot points. Start with the intercepts because they give you two key points right away.

For our equation \( y = \frac{3}{4} x + 600 \), we found the intercepts:

• y-intercept: (0, 600)
• x-intercept: (-800, 0)

Once you have these points, draw them on the coordinate plane. Use a ruler to draw a straight line through both points.

If you need more points to be certain, choose any x-value, plug it into the equation, and solve for y. For example, if x = 200:
\( y = \frac{3}{4} (200) + 600 = 150 + 600 = 750 \)

This gives you another point (200, 750), making your line even more accurate. Plot this new point and draw a line connecting all the points to ensure accuracy.