To find the expression that represents the additive pattern described by the points (3, 16) and (8, 21), we first need to determine the change in y (the output) as we change x (the input) between these two points.
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Identify the coordinates:
- First point: (3, 16) → x₁ = 3, y₁ = 16
- Second point: (8, 21) → x₂ = 8, y₂ = 21
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Calculate the change in x and y:
- Change in x (Δx) = x₂ - x₁ = 8 - 3 = 5
- Change in y (Δy) = y₂ - y₁ = 21 - 16 = 5
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Determine the additive pattern: The change in y is constant (5) for a change of 5 in x, which indicates a linear relationship with a slope of 1.
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Find the linear expression: We can express this pattern in the form \(y = mx + b\) where m is the slope and b is the y-intercept.
Given that:
- Slope \(m = \frac{Δy}{Δx} = 1\)
Using the point (3, 16) to find the y-intercept (b): \[ 16 = 1(3) + b \] \[ 16 = 3 + b \] \[ b = 16 - 3 = 13 \]
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Write the expression: The equation representing the pattern is: \[ y = 1x + 13 \] or simply: \[ y = x + 13 \]
Therefore, the expression that represents the graph is:
x + 13.