Find the Rate of Change: Analyzing y=14+5x Linear Function

Q: Why is the rate of change 5 and not 14?

The rate of change is how much y increases for each unit increase in x. In $ y = 14 + 5x $, the coefficient of x is 5, meaning y increases by 5 for every 1-unit increase in x.

Q: What does the 14 represent in this equation?

The 14 is the y-intercept - it's where the line crosses the y-axis when x = 0. It's not related to the rate of change at all!

Q: Does it matter that the equation is written as y = 14 + 5x instead of y = 5x + 14?

No! Both forms are equivalent. The order of terms doesn't change their meaning. The coefficient of x is still 5 in both cases.

Q: How can I remember which number is the slope?

Think: "The slope is stuck to x!" Whatever number is multiplied by x (the coefficient) is always the slope or rate of change.

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

If you see just x (like in y = x + 3), there's an invisible 1 in front of it. So the slope would be 1, meaning the rate of change is 1.

Slope Identification with Standard Form

Given the linear function:

$y=14+5x$

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 the rate of change for this function.

00:10 First, we'll arrange the equation carefully.

00:14 Now remember, the rate of change is just the slope, which we call M.

00:19 We'll use the equation of a straight line to help us.

00:24 Let's compare and find the slope of our function using this equation.

00:29 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the linear function:

$y=14+5x$

What is the rate of change of the function?

Step-by-step solution

To determine the rate of change of the linear function $y = 14 + 5x$ , we need to identify the structure of the equation. We notice that it is given in the slope-intercept form $y = mx + b$ , where $m$ is the slope or the rate of change.

In the equation $y = 14 + 5x$ , the term involving $x$ is $5x$ . Thus, the coefficient of $x$ , which is $5$ , represents the rate of change or slope of the function.

Therefore, the rate of change of the function is $m = 5$ .

Final Answer

$m=5$

Key Points to Remember

Essential concepts to master this topic

Slope-Intercept Form: In $y = mx + b$ , m equals the slope
Technique: Rewrite $y = 14 + 5x$ as $y = 5x + 14$ to identify slope
Check: Coefficient of x is 5, so rate of change = 5 ✓

Common Mistakes

Avoid these frequent errors

Confusing the constant term with the slope
Don't think the rate of change is 14 because it's the first number = wrong slope! The constant term 14 is the y-intercept, not the rate of change. Always identify the coefficient of x as the slope.

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 rate of change 5 and not 14?

The rate of change is how much y increases for each unit increase in x. In $y = 14 + 5x$ , the coefficient of x is 5, meaning y increases by 5 for every 1-unit increase in x.

What does the 14 represent in this equation?

The 14 is the y-intercept - it's where the line crosses the y-axis when x = 0. It's not related to the rate of change at all!

Does it matter that the equation is written as y = 14 + 5x instead of y = 5x + 14?

No! Both forms are equivalent. The order of terms doesn't change their meaning. The coefficient of x is still 5 in both cases.

How can I remember which number is the slope?

Think: "The slope is stuck to x!" Whatever number is multiplied by x (the coefficient) is always the slope or rate of change.

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

If you see just x (like in y = x + 3), there's an invisible 1 in front of it. So the slope would be 1, meaning the rate of change is 1.

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