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Select Equation Given Set of Points

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Identify expressions and equations

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Q. Select Equation Given Set of Points

Calculate Slope Differences: To find the equation that corresponds to the given set of points, we need to determine if there is a linear relationship between the $x$ and $y$ values. We can start by calculating the differences in $y$ -values and $x$ -values between consecutive points to see if the slope is constant.
Calculate Slope: The slope $m$ between the first two points $(1, 4)$ and $(2, 7)$ is calculated as $\frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 4}{2 - 1} = \frac{3}{1} = 3$ .
Determine Linearity: Now, let's calculate the slope between the second and third points $(2, 7)$ and $(3, 10)$ . The slope is $(y_3 - y_2) / (x_3 - x_2) = (10 - 7) / (3 - 2) = 3 / 1 = 3$ .
Find Y-Intercept: Since the slope is the same between the first two points and the second two points, we can conclude that the points lie on a straight line, and the slope of that line is $3$ .
Solve for Y-Intercept: Next, we need to find the y-intercept $b$ of the line. We can use any of the given points for this. Let's use the first point $(1, 4)$ . The equation of a line is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. Plugging in the values, we get $4 = 3(1) + b$ .
Final Equation: Solving for $b$ , we get $b = 4 - 3 = 1$ .
Final Equation: Solving for $b$ , we get $b = 4 - 3 = 1$ .Now we have both the slope ( $m = 3$ ) and the y-intercept ( $b = 1$ ), so the equation of the line is $y = 3x + 1$ .

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