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Problem 1

# In Exercises $$1-6,$$ a table of values for a linear function is given. Fill in the missing value and calculate $$m$$ in each case. $$\begin{array}{|c|c|c|c|} \hline x & -1 & 0 & 1 \\ \hline y & 5 & 8 & \\\ \hline \end{array}$$

Expert verified
The missing y-value is 11, and the slope (m) of the linear function is 3.
See the step by step solution

## Step 1: Identify Known Points

We are given the points (-1, 5) and (0, 8) from the table.

## Step 2: Calculate the Slope Between the Known Points

Using the slope formula: $$m = \frac{y2 - y1}{x2 - x1}$$, we can calculate the slope between the known points (-1, 5) and (0, 8). $$m = \frac{8-5}{0-(-1)} = \frac{3}{1} = 3$$

## Step 3: Use the Constant Slope to Find the Missing Value

Since the slope for a linear function is constant, we can use the fact that the slope between (-1, 5) and (1, y) should also be 3. Using the slope formula between (-1, 5) and (1, y): $$3 = \frac{y-5}{1-(-1)}$$

## Step 4: Solve for the Missing Value

Now we can solve the equation to find the missing y-value: $$3 = \frac{y-5}{2}$$ Multiplying both sides by 2: $$6 = y - 5$$ Adding 5 to both sides: $$11 = y$$

## Step 5: Fill in the Missing Value and State the Slope

Now we can fill in the missing value in the table and state the slope: $$\begin{array}{|c|c|c|c|} \hline x & -1 & 0 & 1 \\\ \hline y & 5 & 8 & 11\\\ \hline \end{array}$$ The slope (m) of the linear function is 3.

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