Linear Function: Fitting Equations to Tables with Values (-1,10), (0,8), (1,6)

Q: Why is the slope negative when the table shows decreasing y-values?

A negative slope means the line goes downward from left to right. As x increases from -1 to 0 to 1, y decreases from 10 to 8 to 6. This creates a downward trend, which always gives a negative slope!

Q: How do I know which point to use for finding the y-intercept?

The y-intercept occurs when x = 0. In this table, when x = 0, y = 8, so the y-intercept is 8. If x = 0 isn't in your table, use any point with the slope in $ y = mx + b $ to solve for b.

Q: Can I use any two points to find the slope?

Yes! Since it's a linear function, any two points will give you the same slope. Try using (-1, 10) and (1, 6): $ m = \frac{6-10}{1-(-1)} = \frac{-4}{2} = -2 $. Same result!

Q: How do I check if my equation is correct?

Substitute all three points into your equation. For $ y = -2x + 8 $: When x = -1, y = -2(-1) + 8 = 10 ✓. When x = 0, y = -2(0) + 8 = 8 ✓. When x = 1, y = -2(1) + 8 = 6 ✓

Q: What if I get a different equation form?

Linear equations can be written in different forms! $ y = -2x + 8 $ is slope-intercept form. You might also see $ 2x + y = 8 $ (standard form). Both represent the same line!

Linear Functions with Negative Slopes

Given the two tables of values x and and.

These tables represent a linear function. Fit an equation of a linear function to each one.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the algebraic representation for the given table

00:03 Let's take 2 given points

00:11 Use the formula to find the slope using 2 points on the graph

00:17 Substitute appropriate values according to the given data, and solve to find the slope

00:29 This is the graph's slope

00:37 Use the formula to represent a linear function

00:41 Substitute the point and solve for the unknown B

00:52 This is the Y-axis intersection point (unknown B)

00:58 Accordingly substitute the slope and intersection point to find the function

01:07 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

Given the two tables of values x and and.

These tables represent a linear function. Fit an equation of a linear function to each one.

Step-by-step solution

To solve for the linear function, we need to follow these structured steps:

Step 1: Calculate the slope ( $m$ ).

Use the points $(-1, 10)$ and $(0, 8)$ . The slope $m = \frac{8 - 10}{0 - (-1)} = \frac{-2}{1} = -2$ .

Step 2: Determine the y-intercept ( $b$ ).

Use the point-slope form with the point $(0, 8)$ (since $x = 0$ is the y-intercept directly). Thus, $b = 8$ .

Step 3: Write the equation of the line.

The equation fitting the table values is $y = -2x + 8$ .

Therefore, the equation of the linear function is $y = -2x + 8$ .

Final Answer

$y=-2x+8$

Key Points to Remember

Essential concepts to master this topic

Slope Formula: Use any two points: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Technique: Find slope first: $m = \frac{8-10}{0-(-1)} = \frac{-2}{1} = -2$
Check: Substitute all points into final equation: $y = -2x + 8$ ✓

Common Mistakes

Avoid these frequent errors

Mixing up coordinates when calculating slope
Don't switch x and y values or use points in wrong order = wrong slope sign! This gives you a positive slope when it should be negative. Always write points clearly as (x₁, y₁) and (x₂, y₂) and subtract consistently.

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

Why is the slope negative when the table shows decreasing y-values?

A negative slope means the line goes downward from left to right. As x increases from -1 to 0 to 1, y decreases from 10 to 8 to 6. This creates a downward trend, which always gives a negative slope!

How do I know which point to use for finding the y-intercept?

The y-intercept occurs when x = 0. In this table, when x = 0, y = 8, so the y-intercept is 8. If x = 0 isn't in your table, use any point with the slope in $y = mx + b$ to solve for b.

Can I use any two points to find the slope?

Yes! Since it's a linear function, any two points will give you the same slope. Try using (-1, 10) and (1, 6): $m = \frac{6-10}{1-(-1)} = \frac{-4}{2} = -2$ . Same result!

How do I check if my equation is correct?

Substitute all three points into your equation. For $y = -2x + 8$ : When x = -1, y = -2(-1) + 8 = 10 ✓. When x = 0, y = -2(0) + 8 = 8 ✓. When x = 1, y = -2(1) + 8 = 6 ✓

What if I get a different equation form?

Linear equations can be written in different forms! $y = -2x + 8$ is slope-intercept form. You might also see $2x + y = 8$ (standard form). Both represent the same line!

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