Linear Function y=mx+b: Calculate the slope from a table

To solve this problem, we'll proceed with the following steps:

• Step 1: Find the slope $m$.
• Step 2: Use the slope and a point to find the $y$-intercept $b$.
• Step 3: Write the linear equation.

Let's work through each step:

Step 1: Calculate the slope $m$.
Using the points $(0, 5)$ and $(1, 6)$, the slope $m$ is calculated as follows:

$m = \frac{6 - 5}{1 - 0} = \frac{1}{1} = 1$

Step 2: Use the slope ($m = 1$) to find the $y$-intercept $b$.
We know from the point $(0, 5)$ that when $x = 0$, $y = 5$, which directly gives us the $y$-intercept:

$b = 5$

Step 3: Form the equation of the line using $y = mx + b$.
Substitute the found values into the equation:

$y = 1 \cdot x + 5$

Simplifying gives:

$y = x + 5$

Thus, the equation corresponding to the function is $y = x + 5$.

To find the equation of the linear function corresponding to the given table, follow these steps:

• Step 1: Identify Points
  We have three points from the table: $(-2, 4)$, $(-1, 2)$, and $(1, -2)$.
• Step 2: Calculate the Slope
  The slope $m$ can be calculated using any two points. Choosing $(-2, 4)$ and $(-1, 2)$:
  $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 4}{-1 - (-2)} = \frac{-2}{1} = -2$.
• Step 3: Verify Linear Relationship
  Using $(x, y)$ pairs, check another pair such as $(-1, 2)$ and $(1, -2)$:
  $m = \frac{-2 - 2}{1 - (-1)} = \frac{-4}{2} = -2$.
• Step 4: Select the Correct Equation
  The slope is $-2$, and since a linear function typically can be formatted as $y = mx + b$, we can see if $b = 0$ by trying one of the equations $y = -2x$ given in the choices. Let’s check:
  For $x = -2$: $y = -2(-2) = 4$. Matches.
  For $x = -1$: $y = -2(-1) = 2$. Matches.
  For $x = 1$: $y = -2(1) = -2$. Matches.

Therefore, the equation that corresponds to the function is $y = -2x$.

To determine the equation of the linear function from the given table, we'll follow these steps:

• Step 1: Calculate the Slope ($m$)
• Step 2: Find the Y-intercept ($b$)
• Step 3: Formulate the Linear Equation

Let's delve into each step:

Step 1: Calculate the Slope ($m$)
Using two points from the table, $(2, 13)$ and $(1, 10)$, we calculate the slope $m$ using the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 13}{1 - 2} = \frac{-3}{-1} = 3$

Thus, the slope $m$ is 3.

Step 2: Find the Y-intercept ($b$)
The y-intercept $b$ is easily found because it is where $x = 0$. From the table, when $x = 0$, $y = 7$, therefore $b = 7$.

Step 3: Formulate the Linear Equation
Substitute the values of $m$ and $b$ into the linear equation format:

$y = 3x + 7$

Hence, the equation that represents the linear function is $y = 3x + 7$.

Checking against the provided choices, the equation corresponds to choice 2.

To solve for the linear function, we need to follow these structured steps:

Step 1: Calculate the slope ($m$).

Use the points $(-1, 10)$ and $(0, 8)$. The slope $m = \frac{8 - 10}{0 - (-1)} = \frac{-2}{1} = -2$.

Step 2: Determine the y-intercept ($b$).

Use the point-slope form with the point $(0, 8)$ (since $x = 0$ is the y-intercept directly). Thus, $b = 8$.

Step 3: Write the equation of the line.

The equation fitting the table values is $y = -2x + 8$.

Therefore, the equation of the linear function is $y = -2x + 8$.