Describe a Linear Function: Key Identifiers and Characteristics

Q: Why is x = y - 4 considered linear when x and y are mixed up?

Great observation! A linear function can be written in any equivalent form. When you rearrange $ x = y - 4 $ to $ y = x + 4 $, you get the standard y = mx + b form!

Q: How can y = 1 be a linear function if there's no x?

Excellent question! $ y = 1 $ is a horizontal line - still linear! It has slope = 0, so it's technically $ y = 0x + 1 $. Any constant function is linear.

Q: What makes x² terms automatically non-linear?

When graphed, $ x^2 $ terms create curves, not straight lines. Linear means "like a line" - only first-degree variables (x¹, y¹) create straight lines on graphs!

Q: Can I have both x and y in a linear equation?

Absolutely! Equations like $ 2x + 3y = 6 $ are linear. As long as both variables have exponent 1 and aren't multiplied together (like xy), it's linear.

Q: Why do some of these equations look so confusing?

They're testing if you can recognize linear patterns in disguise! Practice rearranging equations into $ y = mx + b $ form - it makes identification much easier.

Q: What if I can't rearrange the equation easily?

Look for the danger signs first: any $ x^2 $, $ y^2 $, xy terms, or fractions with variables in denominators? If none exist, it's likely linear!

Linear Function Identification with Mixed Equations

Which of the following describes a linear function?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Which representations describe a linear function?

00:03 Let's arrange the equation and isolate Y

00:14 Let's compare to the template of a linear function

00:20 The function matches the template, therefore it's linear

00:26 Let's arrange the equation

00:34 Let's compare to the template of a linear function

00:37 The function doesn't match the template, therefore it's not linear

00:42 Let's arrange the equation and isolate Y

00:50 Let's compare to the template of a linear function

00:57 In this case the slope is 0 and the intersection point equals 1

01:01 The function matches the template, therefore it's linear

01:06 Let's arrange the equation and isolate Y

01:12 Let's compare to the template of a linear function

01:15 The function doesn't match the template, therefore it's not linear

01:18 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 of the following describes a linear function?

Step-by-step solution

To solve this problem, we'll examine each expression to verify which represents a linear function:

Option A: $x = y - 4$

This is a linear function. It can be written in the form $y = x + 4$ , which matches the linear form $y = mx + c$ with $m = 1$ and $c = 4$ .

Option B: $x = 3x^2 + 1$

This equation involves a squared term ( $x^2$ ), which means it's not a linear function. Linear functions do not have variables raised to powers other than one.

Option C: $x = x + y - 1$

Rearrange to isolate $y$ :

$y = 1$ . This is a linear equation representing a horizontal line in the xy-plane.

Option D: $x = x^2 + 4 - y$

This equation also involves a squared term ( $x^2$ ), which disqualifies it as a linear function.

Based on this analysis, both Options A and C describe linear functions, and therefore the correct answer is that Answers A and C are correct.

Final Answer

Answers A and C are correct.

Key Points to Remember

Essential concepts to master this topic

Linear Rule: Functions have variables raised only to first power
Technique: Rearrange $x = y - 4$ to get $y = x + 4$
Check: No squared terms like $x^2$ means it's linear ✓

Common Mistakes

Avoid these frequent errors

Confusing equations with squared terms as linear
Don't assume $x = 3x^2 + 1$ is linear because x appears = quadratic function! The $x^2$ term makes it nonlinear. Always check that ALL variables have exponent of 1 only.

Practice Quiz

Test your knowledge with interactive questions

Which statement best describes the graph below?

FAQ

Everything you need to know about this question

Why is x = y - 4 considered linear when x and y are mixed up?

Great observation! A linear function can be written in any equivalent form. When you rearrange $x = y - 4$ to $y = x + 4$ , you get the standard y = mx + b form!

How can y = 1 be a linear function if there's no x?

Excellent question! $y = 1$ is a horizontal line - still linear! It has slope = 0, so it's technically $y = 0x + 1$ . Any constant function is linear.

What makes x² terms automatically non-linear?

When graphed, $x^2$ terms create curves, not straight lines. Linear means "like a line" - only first-degree variables (x¹, y¹) create straight lines on graphs!

Can I have both x and y in a linear equation?

Absolutely! Equations like $2x + 3y = 6$ are linear. As long as both variables have exponent 1 and aren't multiplied together (like xy), it's linear.

Why do some of these equations look so confusing?

They're testing if you can recognize linear patterns in disguise! Practice rearranging equations into $y = mx + b$ form - it makes identification much easier.

What if I can't rearrange the equation easily?

Look for the danger signs first: any $x^2$ , $y^2$ , xy terms, or fractions with variables in denominators? If none exist, it's likely linear!

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