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

Look at the linear function represented in the diagram.

When is the function positive?

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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