Find the Line Equation Through Points (15,36) and (5,16): Two-Point Form

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

No, it doesn't matter! As long as you stay consistent throughout your calculation. Whether you use (15,36) as point 1 or point 2, you'll get the same slope of 2.

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

Absolutely! You can use either (15,36) or (5,16) in the point-slope formula. Both will give you the same final equation: $ y = 2x + 6 $.

Q: How can I double-check my final equation?

Substitute both original points into your equation. For $ y = 2x + 6 $: Point (5,16) gives 16 = 2(5) + 6 = 16 ✓ and point (15,36) gives 36 = 2(15) + 6 = 36 ✓

Two-Point Form with Integer Coordinates

Find the equation of the line passing through the two points $(15,36),(5,16)$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's find the algebraic form of this function.

00:12 First, we use the formula to calculate the slope from two points.

00:18 Each point has an X value and a Y value.

00:26 We'll plot these points, according to our data, and figure out the slope.

00:38 Here is the slope of the line.

00:46 Now, we will use the line equation.

00:51 Plot the points using the data we have.

00:55 Insert the slope, and solve to find where the line meets the Y axis. We call this point B.

01:09 Focus on finding point B, the Y-intercept.

01:13 Here's where the line crosses the Y axis.

01:18 Put the slope and the Y-intercept, point B, into the line equation.

01:29 That's how you solve the problem!

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 $(15,36),(5,16)$

Step-by-step solution

Let's solve the problem to find the equation of the line.

To determine the equation of the line, we first need to calculate the slope $m$ of the line passing through the points $(15, 36)$ and $(5, 16)$ . The formula for the slope is given by:

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

Substituting the given points into the formula:

$m = \frac{16 - 36}{5 - 15} = \frac{-20}{-10} = 2$

Thus, the slope $m$ is 2.

With the slope and one of the points, we can use the point-slope form of the line equation:

$y - y_1 = m(x - x_1)$

We'll use the point $(5, 16)$ :

$y - 16 = 2(x - 5)$

Expanding the equation, we get:

$y - 16 = 2x - 10$

Add 16 to both sides to solve for $y$ :

$y = 2x - 10 + 16$

$y = 2x + 6$

Therefore, the equation of the line is $y = 2x + 6$ .

Final Answer

$y=2x+6$

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 $y - 16 = 2(x - 5)$ using calculated slope
Verification: Check both points: $2(15) + 6 = 36$ and $2(5) + 6 = 16$ ✓

Common Mistakes

Avoid these frequent errors

Mixing up coordinate order when calculating slope
Don't switch x and y values or use different point orders = wrong slope and wrong equation! This gives you the reciprocal or negative of the correct slope. Always label your points clearly as (x₁,y₁) and (x₂,y₂) and stay consistent throughout.

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

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

No, it doesn't matter! As long as you stay consistent throughout your calculation. Whether you use (15,36) as point 1 or point 2, you'll get the same slope of 2.

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

Absolutely! You can use either (15,36) or (5,16) in the point-slope formula. Both will give you the same final equation: $y = 2x + 6$ .

How do I know my slope calculation is correct?

Check that your slope makes sense! Since y increases from 16 to 36 when x increases from 5 to 15, the slope should be positive. A negative slope would mean the line goes downward.

What if I get a negative number when subtracting?

That's normal! Just be careful with your signs. In this problem: $\frac{16-36}{5-15} = \frac{-20}{-10} = 2$ . Two negatives make a positive!

How can I double-check my final equation?

Substitute both original points into your equation. For $y = 2x + 6$ : Point (5,16) gives 16 = 2(5) + 6 = 16 ✓ and point (15,36) gives 36 = 2(15) + 6 = 36 ✓

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