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 $(2,2),(9,16)$

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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 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

Look at the linear function represented in the diagram.

When is the function positive?

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.

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