Linear Equation: Find Line Through (0,9) with Domain x < 8

Q: Why does the line need a negative slope if it should be positive?

The line starts at (0,9) which is already positive! With a negative slope like $ -1\frac{1}{8} $, as x decreases from 0, y increases even more, keeping the line positive for all \( x .

Q: How do I check if y is positive for the entire domain x < 8?

Test a few values! For example, when x = 0: y = 9 ✓, when x = 4: $ y = -1\frac{1}{8}(4) + 9 = 8.5 $ ✓. Since the line is straight, if it's positive at several points, it's positive throughout.

Q: What does the mixed number -1⅛ mean as a slope?

The mixed number $ -1\frac{1}{8} $ equals $ -\frac{9}{8} $ as an improper fraction. This means for every 8 units right, the line goes down 9 units.

Q: Could any of the other answer choices work?

No! Only option D passes through (0,9). Options A and C have y-intercept 8, while option B has y-intercept -9. The point (0,9) immediately eliminates three choices.

Q: What if the domain was x > 8 instead?

Then you'd need a positive slope! Starting at (0,9), a positive slope would make y even larger as x increases beyond 8, keeping the line positive in that domain.

Linear Equations with Domain Restrictions

Choose the equation that represents a straight line that is positive in the domain $8 > x$

and passes through the point $(0,9)$ .

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Choose functions where the given positive domain fits

00:10 Draw the line according to the data

00:16 Positive and negative domains

00:35 The Y-axis intersection point equals the unknown value B

00:49 Use the line equation

00:54 Substitute a point and solve to find the graph's slope

01:04 Isolate the slope M

01:15 This is the function's slope

01:25 Now substitute appropriate values to find the line equation

01:37 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

Choose the equation that represents a straight line that is positive in the domain $8 > x$

and passes through the point $(0,9)$ .

Step-by-step solution

To solve this problem, we'll identify the correct equation of a line that passes through the point $(0, 9)$ and remains positive when $x < 8$ .

Step 1: Identify the y-intercept using the given point $(0, 9)$ . The y-intercept $c$ is 9, leading to a partial equation: $y = mx + 9$ .
Step 2: Determine the appropriate slope $m$ so that the line is positive for $x < 8$ . This means the line should decrease (positive to the right implies negative to the left) as $x$ decreases from 8, requiring a negative slope.
Step 3: Given the provided choices, $y = -1\frac{1}{8}x + 9$ matches these requirements because it incorporates:

The correct y-intercept at 9.
The negative slope, ensuring $y$ is positive for $x < 8$ .

The correct line equation that fulfills these conditions is therefore $y = -1\frac{1}{8}x + 9$ .

Final Answer

$y=-1\frac{1}{8}x+9$

Key Points to Remember

Essential concepts to master this topic

Y-intercept Rule: Use the given point (0,9) to find constant term
Domain Check: Line must be positive when $x < 8$ , requiring negative slope
Verification: Test x = 0 gives y = 9, and y > 0 for all x < 8 ✓

Common Mistakes

Avoid these frequent errors

Choosing positive slope for domain restriction
Don't assume positive slope makes the line positive in the domain = wrong answer! A positive slope starting at (0,9) would make y negative for x < 0. Always check that the line stays positive throughout the entire specified domain x < 8.

Practice Quiz

Test your knowledge with interactive questions

Look at the function shown in the figure.

When is the function positive?

FAQ

Everything you need to know about this question

Why does the line need a negative slope if it should be positive?

The line starts at (0,9) which is already positive! With a negative slope like $-1\frac{1}{8}$ , as x decreases from 0, y increases even more, keeping the line positive for all $x < 8$ .

How do I check if y is positive for the entire domain x < 8?

Test a few values! For example, when x = 0: y = 9 ✓, when x = 4: $y = -1\frac{1}{8}(4) + 9 = 8.5$ ✓. Since the line is straight, if it's positive at several points, it's positive throughout.

What does the mixed number -1⅛ mean as a slope?

The mixed number $-1\frac{1}{8}$ equals $-\frac{9}{8}$ as an improper fraction. This means for every 8 units right, the line goes down 9 units.

Could any of the other answer choices work?

No! Only option D passes through (0,9). Options A and C have y-intercept 8, while option B has y-intercept -9. The point (0,9) immediately eliminates three choices.

What if the domain was x > 8 instead?

Then you'd need a positive slope! Starting at (0,9), a positive slope would make y even larger as x increases beyond 8, keeping the line positive in that domain.

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Linear Equation: Find Line Through (0,9) with Domain x < 8

Linear Equations with Domain Restrictions

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does the line need a negative slope if it should be positive?

How do I check if y is positive for the entire domain x < 8?

What does the mixed number -1⅛ mean as a slope?

Could any of the other answer choices work?

What if the domain was x > 8 instead?

🌟 Unlock Your Math Potential

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