Calculate Slope Between Points (4,1) and (2,5): Two-Point Formula

Q: Why did I get a negative slope?

A negative slope of -2 means the line goes downward from left to right! When you move from (4,1) to (2,5), you go left 2 units and up 4 units, creating a downward slanting line.

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

No! You can start with either point, but you must be consistent. If (4,1) is first, use 4-2 and 1-5. If (2,5) is first, use 2-4 and 5-1. Both give the same answer!

Q: How can I remember the slope formula?

Think "rise over run" - how much you go up (change in y) divided by how much you go across (change in x). The formula $ \frac{y_2-y_1}{x_2-x_1} $ shows this perfectly!

Q: Can I check my slope calculation?

Yes! Switch the point order and recalculate. With (2,5) first: $ \frac{1-5}{4-2} = \frac{-4}{2} = -2 $. Same answer means you're correct!

Slope Formula with Negative Results

Calculate the slope of the line that passes through the points $(4,1),(2,5)$ .

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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

00:03 We will find the slope using 2 points

00:12 We will use the formula to find the slope using 2 points

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

00:45 This is the slope of the graph

00:49 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

Calculate the slope of the line that passes through the points $(4,1),(2,5)$ .

Step-by-step solution

Remember the formula for calculating a slope using points:

Now, replace the data in the formula with our own:

$\frac{(5-1)}{(2-4)}=\frac{4}{-2}=-2$

Final Answer

-2

Key Points to Remember

Essential concepts to master this topic

Formula: Slope = (y₂ - y₁) ÷ (x₂ - x₁) for any two points
Technique: $\frac{5-1}{2-4} = \frac{4}{-2} = -2$ using point order
Check: Switch point order: $\frac{1-5}{4-2} = \frac{-4}{2} = -2$ same result ✓

Common Mistakes

Avoid these frequent errors

Subtracting coordinates in wrong order
Don't mix up which point is first and second like $\frac{5-1}{4-2} = 2$ = wrong answer! This happens when you use y-values from one order but x-values from another order. Always keep the same point order: if (4,1) is first, then subtract 4-2 for x AND 1-5 for y.

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

Why did I get a negative slope?

A negative slope of -2 means the line goes downward from left to right! When you move from (4,1) to (2,5), you go left 2 units and up 4 units, creating a downward slanting line.

Does it matter which point I call (x₁,y₁)?

No! You can start with either point, but you must be consistent. If (4,1) is first, use 4-2 and 1-5. If (2,5) is first, use 2-4 and 5-1. Both give the same answer!

How can I remember the slope formula?

Think "rise over run" - how much you go up (change in y) divided by how much you go across (change in x). The formula $\frac{y_2-y_1}{x_2-x_1}$ shows this perfectly!

What does slope = -2 actually mean?

Slope of -2 means for every 1 unit you move right, the line drops down 2 units. It's like going down a hill that drops 2 feet for every 1 foot forward!

Can I check my slope calculation?

Yes! Switch the point order and recalculate. With (2,5) first: $\frac{1-5}{4-2} = \frac{-4}{2} = -2$ . Same answer means you're correct!

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