Find the Rate of Change: Solving -1/4y + x = 3

Q: Why can't I just use -1/4 as the rate of change?

The coefficient $ -\frac{1}{4} $ is attached to y, not x! Rate of change is the coefficient of x when the equation is in $ y = mx + b $ form.

Q: What does rate of change actually mean?

Rate of change tells you how much y increases for every 1 unit increase in x. In this case, y increases by 4 units for every 1 unit increase in x.

Q: Do I always need to convert to y = mx + b form?

Yes! The slope-intercept form makes it easy to identify the slope (rate of change) as the coefficient of x. Other forms can be confusing.

Q: How do I multiply by -4 correctly?

Multiply every term by -4: $ -4 \times (-\frac{1}{4}y) = y $, $ -4 \times (-x) = 4x $, and $ -4 \times 3 = -12 $.

Q: Can the rate of change be negative?

Absolutely! A negative rate of change means the line slopes downward, while a positive rate of change (like 4) means it slopes upward.

Linear Equations with Fractional Coefficients

For the following straight line equation, state what is the rate of change?

$-\frac{1}{4}y+x=3$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the slope of the graph

00:04 Let's arrange the equation to match the line equation

00:11 Let's isolate Y

00:36 Let's arrange the equation

00:39 The coefficient of X is the slope of the graph, which is the rate of change

00:42 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

For the following straight line equation, state what is the rate of change?

$-\frac{1}{4}y+x=3$

Step-by-step solution

To solve this problem, let's convert the given equation into the standard slope-intercept form, $y = mx + b$ , to find the rate of change:

Step 1: Start with the equation $-\frac{1}{4}y + x = 3$ .
Step 2: Rearrange the equation to solve for $y$ . Subtract $x$ from both sides to get $-\frac{1}{4}y = -x + 3$ .
Step 3: Multiply every term by $-4$ to isolate $y$ . This gives us $y = 4x - 12$ .

Now that the equation is in the form $y = mx + b$ , the slope $m$ is the coefficient of $x$ , which is $4$ .

Therefore, the rate of change for the given straight line equation is $4$ .

Final Answer

$4$

Key Points to Remember

Essential concepts to master this topic

Rate of Change: The slope equals the coefficient of x in y = mx + b
Technique: Multiply by -4 to get y = 4x - 12
Check: Substitute back: 4(0) - 12 = -12, when x = 0, y = -12 ✓

Common Mistakes

Avoid these frequent errors

Confusing rate of change with y-intercept
Don't identify rate of change as -1/4 from the original equation = wrong coefficient! The rate of change is the slope m in y = mx + b form. Always convert to slope-intercept form first to find the coefficient of x.

Practice Quiz

Test your knowledge with interactive questions

Look at the graph below and determine whether the function's rate of change is constant or not:

FAQ

Everything you need to know about this question

Why can't I just use -1/4 as the rate of change?

The coefficient $-\frac{1}{4}$ is attached to y, not x! Rate of change is the coefficient of x when the equation is in $y = mx + b$ form.

What does rate of change actually mean?

Rate of change tells you how much y increases for every 1 unit increase in x. In this case, y increases by 4 units for every 1 unit increase in x.

Do I always need to convert to y = mx + b form?

Yes! The slope-intercept form makes it easy to identify the slope (rate of change) as the coefficient of x. Other forms can be confusing.

How do I multiply by -4 correctly?

Multiply every term by -4: $-4 \times (-\frac{1}{4}y) = y$ , $-4 \times (-x) = 4x$ , and $-4 \times 3 = -12$ .

Can the rate of change be negative?

Absolutely! A negative rate of change means the line slopes downward, while a positive rate of change (like 4) means it slopes upward.

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