Find the Line Through Points (2,2) and (9,16): Coordinate Geometry

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

No, it doesn't matter! You can choose either point as your first point. Just make sure to be consistent - if (2,2) is your first point, use 2 for both x₁ and y₁.

Q: How can I check if my slope is reasonable?

Look at your points: from (2,2) to (9,16), you move 7 units right and 14 units up. Since you go up more than you go right, the slope should be greater than 1, which matches our answer of 2!

Q: What does a slope of 2 actually mean?

A slope of 2 means for every 1 unit you move right, you move 2 units up. It's like climbing stairs where each step forward takes you 2 steps higher!

Q: Can I simplify fractions before calculating?

It's better to substitute first, then simplify. This way you avoid mistakes. In our example: $ \frac{14}{7} = 2 $ is the final step.

Slope Calculation with Two Given Points

The line passes through the points $(2,2),(9,16)$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the slope of the graph

00:04 We'll use the formula to find the slope using 2 points on the graph

00:10 For each point we'll mark X and Y

00:17 We'll substitute appropriate values according to the given data, and solve to find the slope

00:34 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 $(2,2),(9,16)$

Step-by-step solution

To solve this problem, we'll calculate the slope of the line passing through the points $(2, 2)$ and $(9, 16)$ .

Step 1: Identify the coordinates
Step 2: Apply the slope formula
Step 3: Simplify the expression

Let's proceed:

Step 1: The coordinates given are $(x_1, y_1) = (2, 2)$ and $(x_2, y_2) = (9, 16)$ .

Step 2: The slope $m$ of a line through two points is given by:

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

Substituting the coordinates into the formula, we have:

$m = \frac{16 - 2}{9 - 2}$

Step 3: Simplify the expression:

$m = \frac{14}{7}$

$m = 2$

Therefore, the slope of the line is $\mathbf{m = 2}$ .

Final Answer

$m=2$

Key Points to Remember

Essential concepts to master this topic

Formula: Use m = (y₂ - y₁)/(x₂ - x₁) for any two points
Technique: Substitute (2,2) and (9,16): m = (16-2)/(9-2) = 14/7
Check: Verify by counting rise over run on graph: 14 up, 7 right = 2 ✓

Common Mistakes

Avoid these frequent errors

Switching the order of coordinates in subtraction
Don't mix up coordinates like (y₁ - y₂)/(x₁ - x₂) = (2-16)/(2-9) = -14/-7 = 2! While this gives the correct answer by coincidence, it creates bad habits. Always use consistent order: (y₂ - y₁)/(x₂ - x₁).

Practice Quiz

Test your knowledge with interactive questions

What is the solution to the following inequality?

$$ 10x-4\leq-3x-8 $$

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! You can choose either point as your first point. Just make sure to be consistent - if (2,2) is your first point, use 2 for both x₁ and y₁.

What if I get a negative slope?

Negative slopes are completely normal! They just mean the line is going downward from left to right. A positive slope like 2 means the line goes upward.

How can I check if my slope is reasonable?

Look at your points: from (2,2) to (9,16), you move 7 units right and 14 units up. Since you go up more than you go right, the slope should be greater than 1, which matches our answer of 2!

What does a slope of 2 actually mean?

A slope of 2 means for every 1 unit you move right, you move 2 units up. It's like climbing stairs where each step forward takes you 2 steps higher!

Can I simplify fractions before calculating?

It's better to substitute first, then simplify. This way you avoid mistakes. In our example: $\frac{14}{7} = 2$ is the final step.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Linear Functions questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime