Find the Rate of Change in the Linear Equation 3-y=1/4x

Q: What exactly is the 'rate of change' in a linear equation?

The rate of change is the same as the slope! It tells you how much y changes for every 1-unit increase in x. In $ y = -\frac{1}{4}x + 3 $, y decreases by $ \frac{1}{4} $ when x increases by 1.

Q: Why is the rate of change negative in this problem?

The negative sign means the line is decreasing or sloping downward from left to right. As x gets bigger, y gets smaller at a rate of $ \frac{1}{4} $ unit per unit of x.

Q: How do I rearrange 3-y = (1/4)x to slope-intercept form?

Step by step:Start: $ 3 - y = \frac{1}{4}x $Subtract 3: $ -y = \frac{1}{4}x - 3 $Multiply by -1: $ y = -\frac{1}{4}x + 3 $

Q: What's the difference between slope and y-intercept?

In $ y = mx + b $: m is the slope (rate of change) and b is the y-intercept (where line crosses y-axis). For our equation $ y = -\frac{1}{4}x + 3 $, slope = $ -\frac{1}{4} $ and y-intercept = 3.

Linear Equations with Negative Slopes

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

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

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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 linear equation

00:10 Isolate Y

00:25 The coefficient of X is the slope of the graph, which is the rate of change

00:28 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?

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

Step-by-step solution

To solve this problem, we'll follow the approach of rewriting the equation in slope-intercept form:

Step 1: Arrange the equation to isolate $y$ .
Step 2: Identify the coefficient of $x$ as the slope.
Step 3: Match the slope with the provided choices.

Let's work through these steps:
Step 1: Start with the original equation $3-y=\frac{1}{4}x$ . To solve for $y$ , add $y$ to both sides:
$3 = \frac{1}{4}x + y$ .
Subtract $\frac{1}{4}x$ from both sides to isolate $y$ :
$y = -\frac{1}{4}x + 3$ .
Step 2: In the equation $y = -\frac{1}{4}x + 3$ , the coefficient of $x$ is $-\frac{1}{4}$ , which represents the slope or rate of change.
Step 3: Compare the calculated slope $-\frac{1}{4}$ with the given choices. Choice 2, $-\frac{1}{4}$ , is correct.

Therefore, the rate of change for the given line equation is $-\frac{1}{4}$ .

Final Answer

$-\frac{1}{4}$

Key Points to Remember

Essential concepts to master this topic

Slope-Intercept Form: Rearrange equation to y = mx + b format
Technique: From 3-y = $\frac{1}{4}$ x, get y = $-\frac{1}{4}$ x + 3
Check: Rate of change equals coefficient of x: $-\frac{1}{4}$ ✓

Common Mistakes

Avoid these frequent errors

Confusing rate of change with y-intercept
Don't identify 3 as the rate of change = wrong answer! The constant term is the y-intercept, not the slope. Always identify the coefficient of x as the rate of change after converting to y = mx + b form.

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

What exactly is the 'rate of change' in a linear equation?

The rate of change is the same as the slope! It tells you how much y changes for every 1-unit increase in x. In $y = -\frac{1}{4}x + 3$ , y decreases by $\frac{1}{4}$ when x increases by 1.

Why is the rate of change negative in this problem?

The negative sign means the line is decreasing or sloping downward from left to right. As x gets bigger, y gets smaller at a rate of $\frac{1}{4}$ unit per unit of x.

How do I rearrange 3-y = (1/4)x to slope-intercept form?

Step by step:

Start: $3 - y = \frac{1}{4}x$
Subtract 3: $-y = \frac{1}{4}x - 3$
Multiply by -1: $y = -\frac{1}{4}x + 3$

What's the difference between slope and y-intercept?

In $y = mx + b$ : m is the slope (rate of change) and b is the y-intercept (where line crosses y-axis). For our equation $y = -\frac{1}{4}x + 3$ , slope = $-\frac{1}{4}$ and y-intercept = 3.

Can I check my answer by plugging in values?

Absolutely! Pick any x-value, calculate y using your equation, then verify the rate of change. For example: when x = 4, y = 2. When x = 8, y = 1. The change is $\frac{1-2}{8-4} = -\frac{1}{4}$ ✓

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