Question㸚, Write the linear equation that gives the rule for this table. \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 4 & 31 \\ \hline 5 & 37 \\ \hline 6 & 43 \\ \hline 7 & 49 \\ \hline \end{tabular} [䯚] Write your answer as an equation with y first, followed by an equals sign. $\square$

Studdy Solution

STEP 1

1. The relationship between $x$ and $y$ is linear, meaning it can be expressed in the form $y = mx + b$ .
2. The table provides pairs of $x$ and $y$ values that lie on the line described by the equation.
3. We need to determine the slope $m$ and the y-intercept $b$ to write the equation.

STEP 2

1. Calculate the slope $m$ of the line using the given points.
2. Use the slope and one point to solve for the y-intercept $b$ .
3. Write the equation in the form $y = mx + b$ .

STEP 3

To find the slope $m$ , use the formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ :
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Using the points $(4, 31)$ and $(5, 37)$ :
$m = \frac{37 - 31}{5 - 4} = \frac{6}{1} = 6$

STEP 4

Now that we have the slope $m = 6$ , use one of the points to find the y-intercept $b$ . Let's use the point $(4, 31)$ :
Substitute $m = 6$ , $x = 4$ , and $y = 31$ into the equation $y = mx + b$ :
$31 = 6(4) + b$
$31 = 24 + b$
Solve for $b$ :
$b = 31 - 24 = 7$

STEP 5

Now that we have both the slope $m = 6$ and the y-intercept $b = 7$ , we can write the equation of the line:
$y = 6x + 7$

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