Find the Line Equation Through Points (5,-11) and (1,9)

Q: Why does the order of points matter when calculating slope?

The order doesn't matter as long as you're consistent! Whether you use $ \frac{9-(-11)}{1-5} $ or $ \frac{-11-9}{5-1} $, you'll get the same slope: -5.

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

Absolutely! Using (5, -11) or (1, 9) will give you the same final equation. Pick whichever point has simpler numbers to make your calculations easier.

Q: How do I know which answer choice format to pick?

Look at the answer choices first! If they're in standard form (Ax + By = C), convert your equation to match. If they're in slope-intercept form (y = mx + b), solve for y.

Q: How can I verify my final equation is correct?

Substitute both original points into your equation! For y + 5x = 14: Point (1,9) gives 9 + 5(1) = 14 ✓ and point (5,-11) gives -11 + 5(5) = 14 ✓

Line Equations with Two Given Points

Find the equation of the line passing through the two points $(5,-11),(1,9)$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's begin by finding the algebraic form of the function.

00:12 We'll use a handy formula to find the slope using two points.

00:17 Remember: in each point, the first number is the X-value, and the second is the Y-value.

00:25 Let's substitute the points into the formula to find our slope.

00:36 And here we have the slope of the line.

00:42 Next, we will use the line equation.

00:47 Plug in the given point into the equation.

00:51 Now substitute the slope, and calculate to find the intercept, B.

01:07 Let's isolate the intercept, B.

01:11 Here is the intercept point on the Y-axis.

01:18 Finally, put the intercept and slope back into the line equation.

01:36 Let's tidy up the equation.

01:40 And there you have it! That's 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 $(5,-11),(1,9)$

Step-by-step solution

To solve this problem, we will follow these steps:

Step 1: Calculate the slope of the line passing through the points $(5, -11)$ and $(1, 9)$ .
Step 2: Use the calculated slope and one of the points to determine the equation of the line in point-slope form.
Step 3: Convert the equation to standard form and verify with choices.

Step 1: Calculate the Slope
The slope $m$ is given by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - (-11)}{1 - 5} = \frac{9 + 11}{1 - 5} = \frac{20}{-4} = -5$

Step 2: Use Point-Slope Form
We choose point $(1, 9)$ to use in the point-slope formula:

$y - y_1 = m(x - x_1)$
$y - 9 = -5(x - 1)$

Step 3: Simplify to Standard Form
Expand and rearrange the equation:

$y - 9 = -5x + 5$

Bring all terms to one side:

$y + 5x = 14$

The linear equation in standard form is $y + 5x = 14$ , which matches choice 4.

Final Answer

$y+5x=14$

Key Points to Remember

Essential concepts to master this topic

Slope Formula: Use m = (y₂ - y₁)/(x₂ - x₁) for any two points
Point-Slope Method: Apply y - y₁ = m(x - x₁) with calculated slope m = -5
Standard Form Check: Rearrange to Ax + By = C and verify: y + 5x = 14 ✓

Common Mistakes

Avoid these frequent errors

Mixing up coordinates when calculating slope
Don't switch x and y values or use points inconsistently like (y₂ - y₁)/(x₁ - x₂) = wrong slope! This gives the negative reciprocal and completely wrong equation. Always keep coordinates organized: label points clearly and use the same order throughout.

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 the order of points matter when calculating slope?

The order doesn't matter as long as you're consistent! Whether you use $\frac{9-(-11)}{1-5}$ or $\frac{-11-9}{5-1}$ , you'll get the same slope: -5.

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

Absolutely! Using (5, -11) or (1, 9) will give you the same final equation. Pick whichever point has simpler numbers to make your calculations easier.

How do I know which answer choice format to pick?

Look at the answer choices first! If they're in standard form (Ax + By = C), convert your equation to match. If they're in slope-intercept form (y = mx + b), solve for y.

What if I get a positive slope but the line looks like it's going down?

Double-check your coordinate subtraction! A line from (1, 9) to (5, -11) goes down from left to right, so the slope should be negative. Recheck your $y_2 - y_1$ and $x_2 - x_1$ calculations.

How can I verify my final equation is correct?

Substitute both original points into your equation! For y + 5x = 14: Point (1,9) gives 9 + 5(1) = 14 ✓ and point (5,-11) gives -11 + 5(5) = 14 ✓

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