Variation of a Function: Calculate the rate of change from an equation

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

• Step 1: Understand the equation format
• Step 2: Identify the slope from the equation
• Step 3: Match the slope to the given choices

Now, let's work through each step:
Step 1: The equation $y = 5x + 4$ is in the slope-intercept form $y = mx + b$, where $m$ is the slope.
Step 2: From the equation, the slope $m$ is the coefficient of $x$, which is 5. This represents the rate of change for this line.
Step 3: Among the choices, the correct choice representing the rate of change is 5.

Therefore, the solution to the problem is $5$

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

• Step 1: Recognize the standard form of a linear equation.
• Step 2: Identify the coefficient of $x$ as the rate of change.

Now, let's work through each step.
Step 1: Recognize that the given equation $y = 8x - 3$ is in the slope-intercept form $y = mx + b$, where $m$ is the slope.
Step 2: In this equation, the coefficient of $x$ is 8. Therefore, the coefficient $8$ represents the rate of change or the slope of the line.

Thus, the rate of change for the equation $y = 8x - 3$ is $8$, which corresponds to choice 1.

To solve this problem, we need to determine the rate of change of the linear equation given by:

$y = -4x + 3$

The standard form for a linear equation is:

$y = mx + b$

where $m$ represents the slope of the line. The slope indicates the rate of change of the function, which describes how much the value of $y$ changes for a unit change in $x$.

In the equation $y = -4x + 3$, the slope $m$ is $-4$. This means that for every unit increase in $x$, the value of $y$ decreases by 4. Thus, the rate of change for this equation is $-4$.

Therefore, the rate of change is $-4$.

To determine the rate of change for the given line equation $y = \frac{1}{4}x + 8$:

• The equation is in the slope-intercept form, $y = mx + b$, where $m$ represents the slope or the rate of change.
• By comparing the given equation $y = \frac{1}{4}x + 8$ with the general form $y = mx + b$, we see that the slope $m$ is $\frac{1}{4}$.
• The rate of change for this line, thus, is $\frac{1}{4}$.

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

To determine the rate of change for the given line equation, we recognize that the equation $y = -\frac{3}{8} - 4x$ is in the slope-intercept form $y = mx + b$, where $m$ is the slope and represents the rate of change.

In the equation provided, the term $-4x$ indicates that the slope or rate of change is $m = -4$.

Thus, the rate of change of the given straight line is $-4$.

Comparing this to the provided choices, the correct answer is:

• Choice id "2": $-4$

Therefore, the rate of change for the linear equation is $-4$.

To solve this problem, let's follow these steps:

• Step 1: Convert the given equation to the slope-intercept form.
• Step 2: Identify the slope as the rate of change.

Now, let's work through each step:

Step 1: Convert to Slope-Intercept Form
The given equation is $-5x + y = 3$. To convert it to the slope-intercept form, solve for $y$:

Add $5x$ to both sides to isolate $y$:

$y = 5x + 3$

Here, the equation is now in the form $y = mx + b$, where $m$ represents the slope.

Step 2: Identify the Slope
From the equation $y = 5x + 3$, we can see that the slope $m$ is $5$.

Therefore, the rate of change for the given line equation is $5$.

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 $-\frac{1}{4}$.

To determine the rate of change for the given equation $-8y + 2x = 6$, follow these steps:

• Step 1: Rearrange the equation to isolate $y$. Start by subtracting $2x$ from both sides:
  $-8y = -2x + 6$.
• Step 2: Solve for $y$ by dividing every term by $-8$ to get $y$ by itself:
  $y = \frac{-2}{-8}x + \frac{6}{-8}$.
• Step 3: Simplify the fractions:
  $y = \frac{1}{4}x - \frac{3}{4}$.

In the equation $y = \frac{1}{4}x - \frac{3}{4}$, the coefficient of $x$ is $\frac{1}{4}$, which represents the rate of change (slope) of the line. Therefore, the rate of change is $\frac{1}{4}$.

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 $4$.

To solve this problem, we need to determine the rate of change of the given line equation $-x + y = 0$.

Let's proceed step-by-step:

• Step 1: Identify the given equation, which is $-x + y = 0$.
• Step 2: Rearrange the equation to fit the slope-intercept form $y = mx + b$.
  To do this, add $x$ to both sides:
  $y = x$
• Step 3: Identify the slope $m$.
  In the equation $y = x$, the term $x$ has a coefficient of $1$, which means the slope $m = 1$.
• Step 4: Interpret the result.
  The slope $1$ is the rate of change of the line, meaning for every unit increase in $x$, $y$ increases by 1 unit.

Therefore, the rate of change of the line is $1$.