Find the Linear Equation: Line Through Point (3,14) at 135 Degrees

Q: Why is tan(135°) equal to -1?

135° is in the second quadrant where tangent is negative. Since 135° = 180° - 45°, we get $ \tan(135°) = \tan(180° - 45°) = -\tan(45°) = -1 $.

Q: How do I remember which angles give which slopes?

Key angles to memorize: 0° gives slope 0, 45° gives slope 1, 90° is undefined, 135° gives slope -1. These are the most common in problems!

Q: Can I use point-slope form directly?

Absolutely! Once you have the slope m = -1 and point (3,14), use $ y - 14 = -1(x - 3) $ and simplify to get your final equation.

Q: Why does the problem give multiple equation formats as answers?

Linear equations can be written in different forms: slope-intercept ($ y = mx + b $) or standard form ($ Ax + By = C $). They're mathematically equivalent!

Q: What if I get the angle in radians instead of degrees?

Convert first! $ 135° = \frac{3\pi}{4} $ radians. Then use $ \tan(\frac{3\pi}{4}) = -1 $. The slope calculation stays the same.

Linear Equations with Angular Slope

Which algebraic equation represents a straight line that passes through the point $(3,14)$ and creates an angle of 135 degrees with the positive part of the x axis?

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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

00:03 We'll use the formula to calculate slope based on the angle with X-axis

00:07 We'll substitute the angle according to the given data and calculate to find the slope

00:16 This is the line's slope

00:19 Now we'll use the line equation

00:24 We'll substitute the point according to the given data

00:30 We'll substitute the slope and solve to find the intersection point (B)

00:42 We'll isolate the intersection point (B)

00:47 This is the intersection point with the Y-axis

00:57 Now we'll substitute the intersection point and slope in the line equation

01:16 We'll arrange the equation

01:22 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

Which algebraic equation represents a straight line that passes through the point $(3,14)$ and creates an angle of 135 degrees with the positive part of the x axis?

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Calculate the slope
Step 2: Apply the point-slope form to get the line equation
Step 3: Simplify and verify against given choices

Now, let's work through each step:
Step 1: The slope for a line making 135 degrees with the x-axis is $m = \tan(135^\circ) = \tan(135^\circ - 180^\circ) = \tan(-45^\circ) = -1$ .
Step 2: Using the point-slope formula $y - y_1 = m(x - x_1)$ with $(x_1, y_1) = (3, 14)$ and $m = -1$ , we have:
$y - 14 = -1(x - 3)$ .
Simplifying, $y - 14 = -x + 3$ .
Rearranging terms gives $y = -x + 17$ , or equivalently $y + x = 17$ .

Therefore, the equation of the line is $y + x = 17$ .

Final Answer

$y+x=17$

Key Points to Remember

Essential concepts to master this topic

Angle Rule: Convert angle to slope using $m = \tan(\theta)$
Technique: For 135°: $\tan(135°) = -1$ , so slope is -1
Check: Point (3,14) in $y + x = 17$ : 14 + 3 = 17 ✓

Common Mistakes

Avoid these frequent errors

Using the angle directly as slope instead of finding tangent
Don't use 135 as the slope = equation like y = 135x + b! The angle measures rotation from x-axis, not steepness. Always find slope using m = tan(angle) first.

Practice Quiz

Test your knowledge with interactive questions

Find the equation of the line passing through the two points $$ (9,10),(99,100) $$

FAQ

Everything you need to know about this question

Why is tan(135°) equal to -1?

135° is in the second quadrant where tangent is negative. Since 135° = 180° - 45°, we get $\tan(135°) = \tan(180° - 45°) = -\tan(45°) = -1$ .

How do I remember which angles give which slopes?

Key angles to memorize: 0° gives slope 0, 45° gives slope 1, 90° is undefined, 135° gives slope -1. These are the most common in problems!

Can I use point-slope form directly?

Absolutely! Once you have the slope m = -1 and point (3,14), use $y - 14 = -1(x - 3)$ and simplify to get your final equation.

Why does the problem give multiple equation formats as answers?

Linear equations can be written in different forms: slope-intercept ( $y = mx + b$ ) or standard form ( $Ax + By = C$ ). They're mathematically equivalent!

What if I get the angle in radians instead of degrees?

Convert first! $135° = \frac{3\pi}{4}$ radians. Then use $\tan(\frac{3\pi}{4}) = -1$ . The slope calculation stays the same.

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