Find the Line Through Points (5,7) and (1,3): Coordinate Geometry

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

No, it doesn't matter! As long as you're consistent. Whether you use $ \frac{3-7}{1-5} $ or $ \frac{7-3}{5-1} $, both give the same slope value.

Q: What if I get a negative number in both numerator and denominator?

That's great! When you have $ \frac{-4}{-4} $, the negative signs cancel out to give you a positive slope. Remember: negative ÷ negative = positive.

Q: What does a slope of 1 actually mean?

A slope of 1 means the line rises 1 unit up for every 1 unit right. It's a 45-degree angle - perfectly diagonal!

Slope Calculation with Two Given Points

The line passes through the points $(5,7),(1,3)$

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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 At each point mark X and Y

00:15 Use the formula to find the slope using 2 points on the graph

00:21 Substitute appropriate values according to the given data, and solve to find the slope

00:45 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 $(5,7),(1,3)$

Step-by-step solution

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

Identify the coordinates of the points.
Apply the slope formula.
Calculate the slope value.

Let's work through the steps:

We are given two points on a line: $(5,7)$ and $(1,3)$ .

Step 1: Assign the coordinates: $(x_1, y_1) = (5, 7)$ and $(x_2, y_2) = (1, 3)$ .

Step 2: Use the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ .

Substitute the coordinates into the formula:
$m = \frac{3 - 7}{1 - 5} \\ = \frac{-4}{-4} \\ = 1$

Therefore, the slope of the line passing through the points $(5,7)$ and $(1,3)$ is $\bm{m = 1}$ .

Thus, the correct answer is $\bm{m = 1}$ , corresponding to choice 1.

Final Answer

$m=1$

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: Substitute (5,7) and (1,3): $m = \frac{3-7}{1-5} = \frac{-4}{-4} = 1$
Check: Verify slope by counting rise over run on graph: up 4, left 4 gives slope 1 ✓

Common Mistakes

Avoid these frequent errors

Mixing up coordinate order in slope formula
Don't randomly subtract coordinates like $\frac{7-3}{5-1} = \frac{4}{4} = 1$ - you got lucky this time! Inconsistent coordinate pairing usually gives wrong slopes. Always use the same point order: if you pick (5,7) as $(x_1, y_1)$ , then (1,3) must be $(x_2, y_2)$ .

Practice Quiz

Test your knowledge with interactive questions

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 is (x₂, y₂)?

No, it doesn't matter! As long as you're consistent. Whether you use $\frac{3-7}{1-5}$ or $\frac{7-3}{5-1}$ , both give the same slope value.

What if I get a negative number in both numerator and denominator?

That's great! When you have $\frac{-4}{-4}$ , the negative signs cancel out to give you a positive slope. Remember: negative ÷ negative = positive.

How can I tell if my slope makes sense?

Look at the points on a graph! From (1,3) to (5,7), you move right and up, so the slope should be positive. Our answer m = 1 is positive, which makes sense!

What does a slope of 1 actually mean?

A slope of 1 means the line rises 1 unit up for every 1 unit right. It's a 45-degree angle - perfectly diagonal!

Can I check my answer without graphing?

Yes! Pick any point on the line and use the slope to find another point. From (5,7), go right 1 and up 1 to get (6,8). Check if this point would work with the original slope formula.

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