Find the Linear Equation Through Points (-2,-6) and (4,12): Two-Point Form

Q: Why did the y-intercept come out to be zero?

When b = 0, it means the line passes through the origin (0,0). This happens when the line is proportional - notice that $ \frac{y}{x} = 3 $ for both points!

Q: Can I use either point to find the y-intercept?

Yes! Both points will give you the same b-value. Try using (-2,-6): $ -6 = 3(-2) + b $, so $ b = 0 $.

Q: What if I get a negative slope instead?

Check your subtraction! Remember that $ 12-(-6) = 12+6 = 18 $ and $ 4-(-2) = 4+2 = 6 $. Subtracting negatives becomes addition.

Q: Why is the slope positive when one point has negative coordinates?

The slope depends on the direction of the line, not individual coordinates. Since y increases as x increases (from (-2,-6) to (4,12)), the line goes upward = positive slope.

Two-Point Form with Zero Y-Intercept

Find the equation of the line passing through the two points $(-2,-6),(4,12)$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the algebraic representation of the function

00:04 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

00:17 Plot the points according to the data and find the slope

00:30 This is the line's slope

00:35 Now use the line equation

00:39 Plot the point according to the data

00:44 Input the slope, and solve to find the intersection point (B)

00:59 Isolate the intersection point (B)

01:05 This is the intersection point with the Y-axis

01:11 Now input the intersection point and slope into the line equation

01:19 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

Find the equation of the line passing through the two points $(-2,-6),(4,12)$

Step-by-step solution

In the first step, we'll find the slope using the formula:

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

We'll substitute according to the given points:

$m=\frac{12-(-6)}{4-(-2)}$

$m=\frac{18}{6}=3$

Now we'll choose the point (4,12) and use the formula:

$y=mx+b$

$12=3\times4+b$

$12=12+b$

$b=0$

We'll substitute the data into the formula to find the equation of the line:

$y=3x$

Final Answer

$y=3x$

Key Points to Remember

Essential concepts to master this topic

Slope Formula: Use $m=\frac{y_2-y_1}{x_2-x_1}$ with correct point order
Technique: Calculate $m=\frac{12-(-6)}{4-(-2)}=\frac{18}{6}=3$ step by step
Check: Verify both points satisfy $y=3x$ : (-2,-6) and (4,12) ✓

Common Mistakes

Avoid these frequent errors

Mixing up coordinate order when calculating slope
Don't use $m=\frac{4-(-2)}{12-(-6)}=\frac{6}{18}=\frac{1}{3}$ = wrong slope! This swaps x and y differences, giving the reciprocal. Always subtract y-coordinates in numerator and x-coordinates in denominator.

Practice Quiz

Test your knowledge with interactive questions

Look at the function shown in the figure.

When is the function positive?

FAQ

Everything you need to know about this question

Why did the y-intercept come out to be zero?

When b = 0, it means the line passes through the origin (0,0). This happens when the line is proportional - notice that $\frac{y}{x} = 3$ for both points!

Can I use either point to find the y-intercept?

Yes! Both points will give you the same b-value. Try using (-2,-6): $-6 = 3(-2) + b$ , so $b = 0$ .

What if I get a negative slope instead?

Check your subtraction! Remember that $12-(-6) = 12+6 = 18$ and $4-(-2) = 4+2 = 6$ . Subtracting negatives becomes addition.

How do I know which point is (x₁,y₁) and which is (x₂,y₂)?

It doesn't matter! You can choose either point as your starting point. Just be consistent - if (-2,-6) is point 1, then (4,12) must be point 2.

Why is the slope positive when one point has negative coordinates?

The slope depends on the direction of the line, not individual coordinates. Since y increases as x increases (from (-2,-6) to (4,12)), the line goes upward = positive slope.

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