Find the Line Equation Through Points (12,40) and (2,10): Step-by-Step

Q: Why does my slope calculation give a different answer?

Make sure you're consistent with your point labels! If (2,10) is $ (x_1, y_1) $, then (12,40) must be $ (x_2, y_2) $. Mixing up the order gives you the wrong slope.

Q: Can I use either point in the point-slope form?

Yes! You can use either (2,10) or (12,40) in $ y - y_1 = m(x - x_1) $. Both will give you the same final equation when simplified correctly.

Q: How do I know which form of the answer is correct?

All equivalent forms are correct! $ y = 3x + 4 $, $ y - 4 = 3x $, and $ 3x - y = -4 $ are all the same line. Choose the form that matches your answer choices.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the algebraic representation of the function

00:03 We'll use the formula to find the slope using 2 points

00:11 In each point the left number represents X-axis and the right Y

00:16 We'll substitute the points according to the given data and find the slope

00:29 This is the line's slope

00:34 Now we'll use the line equation

00:41 We'll substitute the point according to the given data

00:46 We'll substitute the slope, and solve to find the intersection point (B)

00:57 We'll isolate the intersection point (B)

01:01 This is the intersection point with the Y-axis

01:06 Now we'll substitute the intersection point and slope in the line equation

01:18 We'll arrange the equation

01:23 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Find the equation of the line passing through the two points $(12,40),(2,10)$

2

Step-by-step solution

To find the equation of the line passing through the points $(12,40)$ and $(2,10)$ , follow these steps:

Calculate the slope $m$ .
Use the point-slope form to find the equation.

Step 1: Calculate the slope $m$ .
Using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ , where $(x_1, y_1) = (2, 10)$ and $(x_2, y_2) = (12, 40)$ , we find:

$m = \frac{40 - 10}{12 - 2} = \frac{30}{10} = 3$

Step 2: Use the point-slope form $y - y_1 = m(x - x_1)$ .
We use one of the points, say $(2,10)$ , and the calculated slope $m = 3$ to write:

$y - 10 = 3(x - 2)$

Simplify the equation:
$y - 10 = 3x - 6$

Add 10 to both sides:
$y = 3x - 6 + 10$

$y = 3x + 4$

The line equation is $y = 3x + 4$ . Comparing to the given correct answer and available choices, we note:

Therefore, the expression $y - 4 = 3x$ is equivalent to $y = 3x + 4$ , matching Choice 1, considered correctly due to specific transformation and option alignment.

This equivalent manipulation keeps the original expression correct while aligning with the structural prompt.

Thus, the equation of the line is therefore $y - 4 = 3x$ .

3

Final Answer

$y-4=3x$

Key Points to Remember

Essential concepts to master this topic

Slope Formula: Use $m = \frac{y_2 - y_1}{x_2 - x_1}$ to find rate of change
Point-Slope Method: Apply $y - y_1 = m(x - x_1)$ with (2,10) gives $y - 10 = 3(x - 2)$
Verification: Check both points: (12,40) gives 40 - 4 = 36 and 3(12) = 36 ✓

Common Mistakes

Avoid these frequent errors

Mixing up coordinate order in slope formula
Don't subtract x-coordinates in numerator and y-coordinates in denominator = slope reciprocal error! This gives $\frac{1}{3}$ instead of 3, making your entire equation wrong. Always use $m = \frac{y_2 - y_1}{x_2 - x_1}$ consistently.

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

Why does my slope calculation give a different answer?

+

Make sure you're consistent with your point labels! If (2,10) is $(x_1, y_1)$ , then (12,40) must be $(x_2, y_2)$ . Mixing up the order gives you the wrong slope.

Can I use either point in the point-slope form?

+

Yes! You can use either (2,10) or (12,40) in $y - y_1 = m(x - x_1)$ . Both will give you the same final equation when simplified correctly.

How do I know which form of the answer is correct?

+

All equivalent forms are correct! $y = 3x + 4$ , $y - 4 = 3x$ , and $3x - y = -4$ are all the same line. Choose the form that matches your answer choices.

What if I get a negative slope?

+

Negative slopes are completely normal! They just mean the line goes downward from left to right. Always double-check your subtraction: $y_2 - y_1$ and $x_2 - x_1$ .

Why do we need two points to find a line equation?

+

Two points completely determine a unique line! With just one point, there are infinitely many possible lines passing through it. The second point gives us the direction (slope).

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Linear Functions questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

Find the Line Equation Through Points (12,40) and (2,10): Step-by-Step

Linear Equations with Point-Slope Form

❤️ Continue Your Math Journey!