Find the Rate of Change in the Linear Function y=10-2x

Q: What's the difference between slope and rate of change?

They're the same thing! In linear functions, slope and rate of change both describe how much y changes when x increases by 1. Both terms are used interchangeably in mathematics.

Q: How do I rewrite y = 10 - 2x in standard form?

Rearrange to $ y = -2x + 10 $. Now it clearly matches $ y = mx + b $ where m = -2 (slope) and b = 10 (y-intercept).

Q: Does the order of terms matter when finding the slope?

No! Whether you write $ y = 10 - 2x $ or $ y = -2x + 10 $, the coefficient of x is still -2. The slope doesn't change based on term order.

Q: What if there's no number in front of x?

If you see just x (like in y = x + 3), the coefficient is 1. If you see -x (like in y = 5 - x), the coefficient is -1. The invisible coefficient is always 1!

Linear Functions with Negative Slopes

Given the linear function:

$y=10-2x$

What is the rate of change of the function?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's find how quickly the function changes.

00:09 This rate of change is called the slope of the function.

00:13 First, let's organize our equation.

00:16 We'll use a linear equation. Listen closely!

00:21 By comparing, we'll find the slope of our function.

00:25 And that's how we solve this problem. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the linear function:

$y=10-2x$

What is the rate of change of the function?

Step-by-step solution

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

Step 1: Recognize the given function $y = 10 - 2x$ and compare it to the standard linear form $y = mx + b$ .
Step 2: Identify the coefficient of $x$ which is $-2$ .
Step 3: Understand that this coefficient $-2$ is the slope or rate of change of the function.

Now, let's work through each step:
Step 1: The linear function provided is $y = 10 - 2x$ .
Step 2: Comparing this with the standard linear form $y = mx + b$ , we see that the coefficient of $x$ is $-2$ .
Step 3: Therefore, the rate of change (or the slope) of the function is $m = -2$ .

Thus, the rate of change of the linear function is $m = -2$ .

Final Answer

$m=-2$

Key Points to Remember

Essential concepts to master this topic

Standard Form: In y = mx + b, m is the rate of change
Coefficient Method: For y = 10 - 2x, coefficient of x gives m = -2
Verification: Negative slope means function decreases as x increases ✓

Common Mistakes

Avoid these frequent errors

Confusing the constant term with the slope
Don't identify 10 as the rate of change = wrong answer! The constant term (10) is the y-intercept, not the slope. Always look at the coefficient of x, which is -2 in this case.

Practice Quiz

Test your knowledge with interactive questions

For the function in front of you, the slope is?

FAQ

Everything you need to know about this question

Why is the rate of change negative in this function?

The rate of change is -2 because the coefficient of x is -2. This means for every 1 unit increase in x, y decreases by 2 units. The negative sign indicates the function is decreasing!

What's the difference between slope and rate of change?

They're the same thing! In linear functions, slope and rate of change both describe how much y changes when x increases by 1. Both terms are used interchangeably in mathematics.

How do I rewrite y = 10 - 2x in standard form?

Rearrange to $y = -2x + 10$ . Now it clearly matches $y = mx + b$ where m = -2 (slope) and b = 10 (y-intercept).

Does the order of terms matter when finding the slope?

No! Whether you write $y = 10 - 2x$ or $y = -2x + 10$ , the coefficient of x is still -2. The slope doesn't change based on term order.

What if there's no number in front of x?

If you see just x (like in y = x + 3), the coefficient is 1. If you see -x (like in y = 5 - x), the coefficient is -1. The invisible coefficient is always 1!

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