Calculate the Slopes: Lines Through (4,0) to (0,2) and (0,-6) to (4,0)

Q: Does it matter which point I call (x₁, y₁) and which one (x₂, y₂)?

No, it doesn't matter! As long as you're consistent, you'll get the same slope. Just make sure if you use point A as $ (x_1, y_1) $, then point B must be $ (x_2, y_2) $.

Q: Why did Line I get a negative slope?

Line I goes from $ (4,0) $ to $ (0,2) $, which means as x decreases from 4 to 0, y increases from 0 to 2. This creates a negative slope because the line falls from left to right.

Q: What does a slope of 3/2 actually mean?

A slope of $ \frac{3}{2} $ means for every 2 units you move right, the line goes up 3 units. It's the rise over run ratio that shows how steep the line is.

Q: How can I tell if my slope calculation is reasonable?

Look at your points! If the line goes up from left to right, slope should be positive. If it goes down from left to right, slope should be negative. Use this as a quick sanity check!

Slope Calculations with Two-Point Formula

Two straight lines are drawn with the x axis and the y triangle axis.

The first line passes through the points $(4,0),(0,2)$

The second line passes through the points $(0,-6),(4,0)$

Find the slope of each line.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the slope of each line

00:03 We'll use the formula to find the slope using 2 points

00:09 We'll substitute the points according to the given data and solve for the slope

00:26 This is the slope of the first line

00:35 We'll use the same method to find the slope of the second line

00:46 We'll substitute the points in the formula according to the given data and solve for the slope

01:05 And this is the solution to the question

Step-by-step written solution

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Understand the problem

Two straight lines are drawn with the x axis and the y triangle axis.

The first line passes through the points $(4,0),(0,2)$

The second line passes through the points $(0,-6),(4,0)$

Find the slope of each line.

Step-by-step solution

To determine the slopes of the two lines, we will use the slope formula:

Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Let's start with the first line:

Line I:

The points are $(4, 0)$ and $(0, 2)$ . Applying the slope formula:

$m_1 = \frac{2 - 0}{0 - 4} = \frac{2}{-4} = -\frac{1}{2}$

Thus, the slope of Line I is $-\frac{1}{2}$ .

Now, let's calculate the slope for the second line:

Line II:

The points are $(0, -6)$ and $(4, 0)$ . Applying the slope formula:

$m_2 = \frac{0 - (-6)}{4 - 0} = \frac{6}{4} = \frac{3}{2}$

Therefore, the slope of Line II is $\frac{3}{2}$ .

In conclusion, the slopes are as follows:

$\frac{3}{2} = II \text{ , } -\frac{1}{2} = I$ .

The solution matches choice $\text{Choice 3}$ .

The correct answer to the problem is: $\frac{3}{2} = II, -\frac{1}{2} = I$ .

Final Answer

$\frac{3}{2}=II\text{ , -}\frac{1}{2}=I$

Key Points to Remember

Essential concepts to master this topic

Slope Formula: Use $m = \frac{y_2 - y_1}{x_2 - x_1}$ for any two points
Technique: Line I: $\frac{2-0}{0-4} = -\frac{1}{2}$ , Line II: $\frac{0-(-6)}{4-0} = \frac{3}{2}$
Check: Positive slope rises left to right, negative slope falls left to right ✓

Common Mistakes

Avoid these frequent errors

Confusing the order of coordinates in the slope formula
Don't mix up x and y values like $\frac{x_2-x_1}{y_2-y_1}$ = completely wrong slope! This flips the relationship between rise and run. Always use $\frac{y_2-y_1}{x_2-x_1}$ with y-values in numerator, x-values in denominator.

Practice Quiz

Test your knowledge with interactive questions

Which statement best describes the graph below?

FAQ

Everything you need to know about this question

Does it matter which point I call (x₁, y₁) and which one (x₂, y₂)?

No, it doesn't matter! As long as you're consistent, you'll get the same slope. Just make sure if you use point A as $(x_1, y_1)$ , then point B must be $(x_2, y_2)$ .

Why did Line I get a negative slope?

Line I goes from $(4,0)$ to $(0,2)$ , which means as x decreases from 4 to 0, y increases from 0 to 2. This creates a negative slope because the line falls from left to right.

What does a slope of 3/2 actually mean?

A slope of $\frac{3}{2}$ means for every 2 units you move right, the line goes up 3 units. It's the rise over run ratio that shows how steep the line is.

How can I tell if my slope calculation is reasonable?

Look at your points! If the line goes up from left to right, slope should be positive. If it goes down from left to right, slope should be negative. Use this as a quick sanity check!

What if I get a whole number instead of a fraction?

That's perfectly normal! A whole number like 3 is the same as $\frac{3}{1}$ . It just means the line rises 3 units for every 1 unit to the right - a pretty steep line!

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