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

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 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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