Find the Line Through Points (3,7) and (6,14): Coordinate Geometry

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 with your choice. If you pick (3,7) as point 1, then (6,14) must be point 2, and vice versa.

Q: How do I convert the improper fraction 7/3 to a mixed number?

Divide: 7 ÷ 3 = 2 remainder 1. So $ \frac{7}{3} = 2\frac{1}{3} $. The whole number is the quotient, and the remainder over divisor becomes the fraction part.

Q: Can I check my slope calculation another way?

Yes! Count the rise and run on a graph, or use the slope formula backwards. From (3,7) to (6,14): go right 3, up 7, so slope = $ \frac{7}{3} $.

Slope Calculation with Two Given Points

The line passes through the points $(3,7),(6,14)$

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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

00:05 For each point, we'll mark X and Y

00:14 We'll use the formula to find the slope using 2 points on the graph

00:28 We'll substitute appropriate values according to the given data and solve to find the slope

00:49 We'll break down the fraction into a whole number and remainder

00:53 We'll convert whole fraction to whole number and combine into mixed fraction

00:57 And this is the solution to the question

Step-by-step written solution

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Understand the problem

The line passes through the points $(3,7),(6,14)$

Step-by-step solution

To solve this problem, we'll calculate the slope of the line passing through the points $(3, 7)$ and $(6, 14)$ . The formula for the slope $m$ of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

First, we identify our points as follows:
Point 1: $(x_1, y_1) = (3, 7)$
Point 2: $(x_2, y_2) = (6, 14)$

Next, apply the formula:
$x_1 = 3 \\ y_1 = 7 \\ x_2 = 6 \\ y_2 = 14 \\$
Substitute into the slope formula:
$m = \frac{14 - 7}{6 - 3} = \frac{7}{3}$

Therefore, the slope of the line is $m = \frac{7}{3} = 2\frac{1}{3}$ .

The correct choice from the given options is: $m=2\frac{1}{3}$ .

Final Answer

$m=2\frac{1}{3}$

Key Points to Remember

Essential concepts to master this topic

Formula: Use m = (y₂ - y₁)/(x₂ - x₁) for slope between points
Technique: Subtract coordinates: (14-7)/(6-3) = 7/3 = 2⅓
Check: Rise of 7 over run of 3 gives 2⅓ units up per unit right ✓

Common Mistakes

Avoid these frequent errors

Switching the order of coordinates in subtraction
Don't subtract (x₁ - x₂)/(y₁ - y₂) or mix up coordinates = wrong slope direction! This gives you the negative reciprocal or completely wrong value. Always keep the same order: (y₂ - y₁) in numerator and (x₂ - x₁) in denominator.

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! As long as you're consistent with your choice. If you pick (3,7) as point 1, then (6,14) must be point 2, and vice versa.

How do I convert the improper fraction 7/3 to a mixed number?

Divide: 7 ÷ 3 = 2 remainder 1. So $\frac{7}{3} = 2\frac{1}{3}$ . The whole number is the quotient, and the remainder over divisor becomes the fraction part.

What does a slope of 2⅓ actually mean?

It means for every 1 unit you move to the right, the line goes up $2\frac{1}{3}$ units. The line is rising steeply from left to right since the slope is positive and greater than 1.

Can I check my slope calculation another way?

Yes! Count the rise and run on a graph, or use the slope formula backwards. From (3,7) to (6,14): go right 3, up 7, so slope = $\frac{7}{3}$ .

Why is my answer different from the other multiple choice options?

Double-check your arithmetic! Common errors include: wrong subtraction order, calculation mistakes, or incorrect fraction-to-mixed-number conversion. The correct slope $\frac{7}{3} = 2\frac{1}{3}$ .

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