Linear Function y=mx+b: Understanding the concept of slope

To determine the characteristic of the function $y = 2 - 3x$, we will evaluate the slope:

• The given function is in the form $y = mx + b$, which indicates a linear equation. Here, $y = 2 - 3x$ can be rearranged as $y = -3x + 2$, showing that $m = -3$.
• The slope $m$ is $-3$.
• In a linear function, the sign of the slope $m$ determines the function's behavior:
• If the slope $m$ is positive ($m > 0$), the function is increasing.
• If the slope $m$ is negative ($m < 0$), the function is decreasing.
• If the slope $m = 0$, the function is constant.
• Since $m = -3$, which is negative, we conclude that the function is decreasing.

Therefore, the function described by $y = 2 - 3x$ is decreasing.

To determine the equation of the given linear function, follow these steps:

• Step 1: Identify the key information: the slope $m = -3$ and the point $(-6, -3)$.
• Step 2: Use the point-slope form, $y - y_1 = m(x - x_1)$.
• Step 3: Substitute the values $m = -3$, $x_1 = -6$, and $y_1 = -3$ into the formula.
• Step 4: Solve for $y$ to convert to slope-intercept form $y = mx + b$.

Now, let's go through the process:

Use the point-slope form:

$y - (-3) = -3(x - (-6))$

Simplify the equation:

$y + 3 = -3(x + 6)$

Distribute the slope on the right side:

$y + 3 = -3x - 18$

Subtract 3 from both sides to solve for $y$:

$y = -3x - 18 - 3$

which simplifies to:

$y = -3x - 21$

This equation, $y = -3x - 21$, matches the first choice in the provided options.

Therefore, the equation that represents the function is $y = -3x - 21$.

To solve for the equation of a line given a slope and a point:

• Step 1: Use the point-slope form of the linear equation: $y - y_1 = m(x - x_1)$.
• Step 2: Substitute the given slope $m = 5$ and point $(2, 4)$ into the equation.
• Step 3: Simplify and rearrange to convert into slope-intercept form $y = mx + b$.

Substituting into the point-slope form, we have:

$y - 4 = 5(x - 2)$

Distribute the 5 across the terms in the parentheses:

$y - 4 = 5x - 10$

Add 4 to both sides to solve for $y$:

$y = 5x - 10 + 4$

This simplifies to:

$y = 5x - 6$

Therefore, the equation of the line is $y = 5x - 6$.

The correct choice among the options given is $y = 5x - 6$.

To solve the problem of finding the equation of the linear function, we will use the point-slope form, which is:

Step-by-step:

• Step 1: Identify given information: The slope $m = 6$ and the point $(x_1, y_1) = (1, 1)$.

• Step 2: Substitute the slope and point into the point-slope form:

  $y - 1 = 6(x - 1)$

• Step 3: Simplify the equation:

  $y - 1 = 6x - 6$

• Step 4: Solve for $y$ to express in slope-intercept form $y = mx + b$:

  $y = 6x - 6 + 1$

• Step 5: Simplify the right-hand side:

  $y = 6x - 5$

Thus, the equation of the linear function is $y = 6x - 5$.

To solve the problem, we will determine the equation of a line using the point-slope form. The general formula for a line in point-slope form is:

$y - y_1 = m(x - x_1)$

Given the slope $m = 6$ and the point $(1, -4)$, we can substitute these values into the formula:

$y - (-4) = 6(x - 1)$

Simplifying the equation, we get:

$y + 4 = 6x - 6$

Now, we want to express this in slope-intercept form $y = mx + b$. So, we solve for $y$:

$y = 6x - 6 - 4$

Finally, combining like terms gives us the equation:

$y = 6x - 10$

Therefore, the equation that represents the function is $y = 6x - 10$.

The correct answer choice is:

: $y=6x-10$

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

• Step 1: Use the point-slope form with the given point and slope.

• Step 2: Simplify the equation to slope-intercept form.

• Step 3: Identify the correct equation from the options.

Now, let's work through each step:

Step 1: Apply the point-slope form formula:
Given the point $(6, 13)$ and the slope $m = 1$, the point-slope form is:
$y - 13 = 1(x - 6)$

Step 2: Simplify to get the equation in slope-intercept form:
$y - 13 = 1 \cdot (x - 6) \\ y - 13 = x - 6 \\ y = x - 6 + 13 \\ y = x + 7$

Step 3: Compare to find the correct answer:
From the simplified equation $y = x + 7$, the correct choice is:

$y=x+7$

Therefore, the equation representing the function is $y = x + 7$.

Remember the formula for calculating a slope using points:

Now, replace the data in the formula with our own:

$\frac{(5-1)}{(2-4)}=\frac{4}{-2}=-2$

To solve the problem, remember the formula to find the slope using two points

Now, we replace the given points in the calculation:

 $\frac{(0-2)}{(0-(-8)}=\frac{-2}{8}=-\frac{1}{4}$

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

• Step 1: Identify the coordinates of the given points.

• Step 2: Substitute these values into the slope formula.

• Step 3: Simplify to find the slope.

Now, let's work through each step:
Step 1: Given points are $(-6, 1)$ and $(2, 4)$. Thus, we have:
$x_1 = -6, y_1 = 1$, $x_2 = 2, y_2 = 4$.
Step 2: Apply the slope formula:
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 1}{2 - (-6)}$
Step 3: Simplify:
Calculate the numerator: $4 - 1 = 3$.
Calculate the denominator: $2 - (-6) = 2 + 6 = 8$.
Thus, the slope $m$ is:
$m = \frac{3}{8}$

Therefore, the solution to the problem is $\frac{3}{8}$.

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

• Step 1: Identify the given information about the points.
• Step 2: Use the slope formula to find the slope.
• Step 3: Calculate and simplify.

Now, let's compute the slope:

Step 1: The points given are $(0, 4)$ and $(-5, 6)$.

Step 2: Apply the slope formula:

The slope $m$ is given by:

Substitute the known values:

Step 3: Simplify the expression:

Thus, the slope of the line passing through the points $(0, 4)$ and $(-5, 6)$ is $-\frac{2}{5}$.

Therefore, the solution to the problem is $-\frac{2}{5}$.