A table of values of a linear function is shown below. Find the output when the input is n. Type your answer in the space provided.
Understand the Problem
The question is asking for the output of a linear function based on given inputs and outputs in a table format. The output for input 'n' needs to be determined using the existing pattern in the table.
Answer
The output when the input is \( n \) is 23.
Answer for screen readers
The output when the input is ( n ) is 23.
Steps to Solve
- Identify the pattern in the output values
The output values corresponding to the inputs are:
- For input 1, output = 11
- For input 2, output = 14
- For input 3, output = 17
- For input 4, output = 20
- Find the change in output values
Calculate the difference between consecutive output values:
- From 11 to 14: ( 14 - 11 = 3 )
- From 14 to 17: ( 17 - 14 = 3 )
- From 17 to 20: ( 20 - 17 = 3 )
The output increases by 3 for each increase in the input.
- Establish the formula for the linear function
The relationship between the input ( x ) and output ( y ) can be defined as:
$$ y = 3x + b $$
To find ( b ), use one of the points, for example, when ( x = 1 ) and ( y = 11 ):
$$ 11 = 3(1) + b \Rightarrow b = 11 - 3 = 8 $$
So the function is:
$$ y = 3x + 8 $$
- Calculate the output when the input is ( n )
To find the output when the input is ( n ):
$$ y = 3n + 8 $$
- Find the specific output for ( n = 5 )
Substituting ( n = 5 ) into the equation:
$$ y = 3(5) + 8 = 15 + 8 = 23 $$
The output when the input is ( n ) is 23.
More Information
This problem illustrates how linear functions establish a constant rate of change, which can be visualized through a table of inputs and outputs. The formula for the linear function derived from the given points helps predict outputs for any input value.
Tips
- Not recognizing the consistent change in output values, which can lead to incorrect calculations of the linear relationship.
- Forgetting to substitute the correct value into the derived formula when calculating the output.
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