Solve Linear Equation: Finding Perpendicular Line Through (0,0) to y-2x=2y

Q: What does 'negative reciprocal' actually mean?

The negative reciprocal means you flip the fraction and change the sign. For slope -2, flip to get -1/2, then change sign to get $ \frac{1}{2} $!

Q: How do I simplify y - 2x = 2y correctly?

Subtract y from both sides: $ y - 2x - y = 2y - y $, which gives $ -2x = y $. So $ y = -2x $ and the slope is -2.

Q: What if I get confused about which slope to use?

Remember: First find the slope of the given line, then take its negative reciprocal for the perpendicular line. Don't mix them up!

Perpendicular Lines with Negative Reciprocal Slopes

Which function represents a straight line that passes through the point $(0,0)$ and is perpendicular to the line $y-2x=2y$ ?

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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 Isolate Y

00:15 This is the equation of the perpendicular function

00:22 This is the slope of the perpendicular function

00:26 The product of slopes of perpendicular lines is (-1)

00:34 Let's substitute the slope and solve to find the second slope

00:47 This is the slope of our function

00:54 A point through which our function passes according to the given data

00:57 We'll use the line equation

01:04 Let's substitute the point according to the given data

01:08 Let's substitute the slope and solve to find the intersection point (B)

01:17 This is the intersection point with the Y-axis

01:21 Now let's substitute the intersection point and slope in the line equation

01:32 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 function represents a straight line that passes through the point $(0,0)$ and is perpendicular to the line $y-2x=2y$ ?

Step-by-step solution

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

Simplify the given equation to find its slope.
Determine the slope for the perpendicular line using the negative reciprocal.
Write the equation of the line through the origin with this slope.

Let's work through each step:
Step 1: The original line is given by the equation $y - 2x = 2y$ . Simplifying, we subtract $y$ from both sides to get $-2x = y$ or, equivalently, $y = -2x$ . Here, the slope $m_1 = -2$ .

Step 2: For the line to be perpendicular, its slope $m_2$ must satisfy $m_1 \cdot m_2 = -1$ . Thus, we have $-2 \cdot m_2 = -1$ , yielding $m_2 = \frac{1}{2}$ .

Step 3: The equation of the line with this slope that passes through the origin is of the form $y = mx$ . Substituting the slope $\frac{1}{2}$ , we have $y = \frac{1}{2}x$ .

Therefore, the function representing the straight line through the origin and perpendicular to the given line is $y = \frac{1}{2}x$ , which corresponds to choice 1.

Final Answer

$y=\frac{1}{2}x$

Key Points to Remember

Essential concepts to master this topic

Rule: Perpendicular lines have slopes that multiply to equal -1
Technique: From y - 2x = 2y, subtract y to get y = -2x
Check: Slopes -2 and 1/2 multiply: (-2)(1/2) = -1 ✓

Common Mistakes

Avoid these frequent errors

Using the same slope instead of negative reciprocal
Don't just copy the slope -2 from y = -2x for the perpendicular line = parallel lines instead! This creates parallel lines, not perpendicular ones. Always flip the fraction and change the sign to get the negative reciprocal.

Practice Quiz

Test your knowledge with interactive questions

What is the solution to the following inequality?

$$ 10x-4\leq-3x-8 $$

FAQ

Everything you need to know about this question

What does 'negative reciprocal' actually mean?

The negative reciprocal means you flip the fraction and change the sign. For slope -2, flip to get -1/2, then change sign to get $\frac{1}{2}$ !

How do I simplify y - 2x = 2y correctly?

Subtract y from both sides: $y - 2x - y = 2y - y$ , which gives $-2x = y$ . So $y = -2x$ and the slope is -2.

Why does the line pass through (0,0)?

When a line passes through the origin (0,0), its equation has no y-intercept term. So it's just $y = mx$ where m is the slope.

What if I get confused about which slope to use?

Remember: First find the slope of the given line, then take its negative reciprocal for the perpendicular line. Don't mix them up!

How can I double-check my perpendicular slope?

Multiply your two slopes together. If you get -1, they're perpendicular! Like $(-2) \times \frac{1}{2} = -1$ ✓

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