Line Through Points (6,19) and (12,20): Coordinate Geometry Problem

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

No, it doesn't matter as long as you're consistent! Whether you use $ \frac{20-19}{12-6} $ or $ \frac{19-20}{6-12} $, both give $ \frac{1}{6} $.

Q: Why did we get a fraction instead of a whole number?

Slopes can be any real number! A slope of $ \frac{1}{6} $ means the line rises 1 unit for every 6 units it moves right - that's a very gentle upward slope.

Q: How can I visualize what this slope means?

Think of slope as steepness! $ \frac{1}{6} $ means for every 6 steps right, you go up 1 step. It's a very gradual incline, like a gentle ramp.

Q: What if I got a negative slope?

A negative slope means the line goes downward from left to right. Check your subtraction - make sure you're using the same order for both x and y coordinates.

Q: Can I simplify the fraction 1/6 further?

No, $ \frac{1}{6} $ is already in lowest terms since 1 and 6 share no common factors other than 1. This is your final answer!

Slope Formula with Coordinate Points

The line passes through the points $(6,19),(12,20)$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the slope of the graph

00:04 For each point, we'll mark X and Y

00:13 We'll use the formula to find the slope using 2 points on the graph

00:21 We'll substitute appropriate values according to the given data, and solve to find the slope

00:40 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The line passes through the points $(6,19),(12,20)$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the coordinates of the given points.
Step 2: Apply the slope formula.
Step 3: Perform the subtraction and division required by the formula.

Now, let's work through each step:
Step 1: We have the points $(x_1, y_1) = (6, 19)$ and $(x_2, y_2) = (12, 20)$ .
Step 2: The formula for the slope $m$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
Step 3: Substituting the values, we get $m = \frac{20 - 19}{12 - 6} = \frac{1}{6}$ .

Therefore, the slope of the line that passes through the points is $m = \frac{1}{6}$ .

Final Answer

$m=\frac{1}{6}$

Key Points to Remember

Essential concepts to master this topic

Formula: Slope equals change in y divided by change in x
Technique: Use $m = \frac{y_2 - y_1}{x_2 - x_1}$ with ordered subtraction
Check: Verify coordinates are correctly identified as (x₁,y₁) and (x₂,y₂) ✓

Common Mistakes

Avoid these frequent errors

Mixing up coordinate order in subtraction
Don't subtract y₁ - y₂ in numerator while doing x₂ - x₁ in denominator = wrong slope! This creates inconsistent ordering and flips the sign. Always use the same order: (y₂ - y₁) over (x₂ - x₁).

Practice Quiz

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For the function in front of you, the slope is?

FAQ

Everything you need to know about this question

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

No, it doesn't matter as long as you're consistent! Whether you use $\frac{20-19}{12-6}$ or $\frac{19-20}{6-12}$ , both give $\frac{1}{6}$ .

Why did we get a fraction instead of a whole number?

Slopes can be any real number! A slope of $\frac{1}{6}$ means the line rises 1 unit for every 6 units it moves right - that's a very gentle upward slope.

How can I visualize what this slope means?

Think of slope as steepness! $\frac{1}{6}$ means for every 6 steps right, you go up 1 step. It's a very gradual incline, like a gentle ramp.

What if I got a negative slope?

A negative slope means the line goes downward from left to right. Check your subtraction - make sure you're using the same order for both x and y coordinates.

Can I simplify the fraction 1/6 further?

No, $\frac{1}{6}$ is already in lowest terms since 1 and 6 share no common factors other than 1. This is your final answer!

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