Math Problem Statement
Find the relationship between Input and Output in the given table: Input: [1, 2, 3, 4, 5], Output: [8, 10, 12, 14, 16].
Solution
The table you provided shows the relationship between Input and Output values:
Input | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
Output | 8 | 10 | 12 | 14 | 16 |
Step 1: Observe the pattern
- The Output increases by 2 each time as the Input increases by 1.
This indicates a linear relationship.
Step 2: Derive the formula
A linear relationship can be written as:
where is the slope and is the y-intercept.
-
Find the slope :
The slope is calculated as: -
Find the intercept :
Substitute one pair of values, such as , into the equation:
Solve for :
Final Equation:
The relationship between Input and Output is:
Verification:
Input | Equation | Output |
---|---|---|
1 | 8 | |
2 | 10 | |
3 | 12 | |
4 | 14 | |
5 | 16 |
Let me know if you'd like me to explain any step in more detail!
Here are 5 related questions to build on this concept:
- What is the slope of a line that passes through the points and ?
- How do you determine the equation of a line given any two points?
- What would the output be if the input is using this equation?
- How can you graph this equation on a coordinate plane?
- If the output is , what is the corresponding input?
Tip:
For linear relationships, the slope indicates how much the output changes per unit increase in the input.
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Slope and Intercept
Patterns in Input-Output Relationships
Formulas
y = mx + b
Slope: m = (y2 - y1) / (x2 - x1)
Theorems
Slope-Intercept Form
Suitable Grade Level
Grade 6-8