Identify Linear Functions Among Given Equations: Which Are Parallel?

Q: How do I know if an equation is linear?

A linear equation has no exponents greater than 1 on variables. Look for the form y = mx + b. If you see $ x^2 $, $ x^3 $, or fractions with x in denominators, it's not linear!

Q: What makes two lines parallel?

Parallel lines have identical slopes but different y-intercepts. If slopes are the same and y-intercepts are also the same, the lines are identical (same line).

Q: Do I need to expand expressions like ½(x+2)?

Yes, always expand first! $ \frac{1}{2}(x+2) $ becomes $ \frac{1}{2}x + 1 $. Only then can you clearly see the slope and y-intercept to make comparisons.

Q: What if one equation has x² in it?

That equation is quadratic, not linear. Since we need both equations to be linear for parallel lines, any pair with a quadratic equation cannot represent parallel lines.

Q: Can I rearrange equations to standard form instead?

You can, but slope-intercept form (y = mx + b) makes it much easier to compare slopes directly. Stick with this form for parallel line problems.

Linear Functions with Parallel Line Identification

Which of the following represent linear functions and parallel lines?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Choose the functions that are linear and parallel

00:03 Linear function with slope of one-half

00:07 Open parentheses properly, multiply by each factor

00:11 Linear function with slope of one-half

00:17 Functions are parallel when their slopes are equal, therefore they match

00:22 Open parentheses properly, multiply by each factor

00:25 Linear function with slope of 3

00:28 This function is not linear because X is squared

00:41 Linear function with slope of 12

00:47 Linear function with slope of 1

00:52 Functions are parallel when their slopes are equal, therefore they don't match

00:55 Linear function with slope of 3

00:58 Linear function with slope of 2

01:01 Functions are parallel when their slopes are equal, therefore they don't match

01:03 And this is the solution to the question

Step-by-step written solution

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Understand the problem

Which of the following represent linear functions and parallel lines?

Step-by-step solution

To solve this problem, we'll analyze each pair of given equations to see if they are linear and parallel.

Let's examine each pair:

Choice 1:
$y = \frac{1}{2}x + 10$
$y = \frac{1}{2}(x + 2)$ simplifies to $y = \frac{1}{2}x + 1$
Both equations are linear with the same slope of $\frac{1}{2}$ , indicating they are parallel.
Choice 2:
$y = 3(x + 4)$ simplifies to $y = 3x + 12$
$y = 3x^2 + 12$ is not in the form $y = mx + b$ as it includes an $x^2$ term. Thus, it is non-linear.
Choice 3:
$y = 5 + 12x$ is already in the form $y = mx + b$ with $m = 12$
$y = 5 + 12 + x$ simplifies to $y = x + 17$ , which has a slope of 1.
Slopes are different, so not parallel.
Choice 4:
$y = 3x + 2$ , slope $m = 3$
$y = 2x + 3$ , slope $m = 2$
Different slopes, thus not parallel.

Therefore, based on our analysis, the correct choice is Choice 1:

$y = \frac{1}{2}x + 10$ and $y = \frac{1}{2}(x + 2)$

Final Answer

_{$y=\frac{1}{2}x+10$}

_{$y=\frac{1}{2}(x+2)$}

Key Points to Remember

Essential concepts to master this topic

Linear Form: All equations must be in y = mx + b format
Slope Comparison: Expand $\frac{1}{2}(x+2)$ to $\frac{1}{2}x + 1$
Verification: Both slopes equal $\frac{1}{2}$ with different y-intercepts confirms parallel ✓

Common Mistakes

Avoid these frequent errors

Assuming equations are parallel without checking slope equality
Don't just look at similar-looking equations like y = 3x + 2 and y = 2x + 3 = they're not parallel! Different slopes (3 ≠ 2) mean different directions. Always expand all expressions first, then compare slopes exactly.

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

How do I know if an equation is linear?

A linear equation has no exponents greater than 1 on variables. Look for the form y = mx + b. If you see $x^2$ , $x^3$ , or fractions with x in denominators, it's not linear!

What makes two lines parallel?

Parallel lines have identical slopes but different y-intercepts. If slopes are the same and y-intercepts are also the same, the lines are identical (same line).

Do I need to expand expressions like ½(x+2)?

Yes, always expand first! $\frac{1}{2}(x+2)$ becomes $\frac{1}{2}x + 1$ . Only then can you clearly see the slope and y-intercept to make comparisons.

What if one equation has x² in it?

That equation is quadratic, not linear. Since we need both equations to be linear for parallel lines, any pair with a quadratic equation cannot represent parallel lines.

Can I rearrange equations to standard form instead?

You can, but slope-intercept form (y = mx + b) makes it much easier to compare slopes directly. Stick with this form for parallel line problems.

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