Question

fill in the missing numbers to complete the linear equation that gives the rule for this table.
x | y
0 | -37
1 | -40
2 | -43
3 | -46
y = \square x - \square
submit

Explanation:

Step1: Find the slope (m)

The slope \( m \) of a line is calculated by the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Using the points \((0, - 37)\) and \((1, - 40)\), we have \( m=\frac{-40-(-37)}{1 - 0}=\frac{-40 + 37}{1}=\frac{-3}{1}=-3 \).

Step2: Find the y - intercept (b)

The linear equation is in the form \( y=mx + b \). When \( x = 0 \), from the table, \( y=-37 \). Substituting \( x = 0 \), \( y=-37 \) and \( m=-3 \) into \( y=mx + b \), we get \( -37=-3\times0 + b \), so \( b=-37 \). The equation is \( y=-3x-(-37)? \) Wait, no. Wait, the given form is \( y=\square x-\square \). Let's re - express \( y=-3x-37 \)? Wait, no. Wait, when \( x = 0 \), \( y=-37 \), and the slope is - 3. So the equation is \( y=-3x-37 \)? Wait, let's check with \( x = 1 \): \( y=-3(1)-37=-3 - 37=-40 \), which matches. For \( x = 2 \): \( y=-3(2)-37=-6 - 37=-43 \), which matches. For \( x = 3 \): \( y=-3(3)-37=-9 - 37=-46 \), which matches. Wait, the given form is \( y=\square x-\square \). So the first blank is - 3? No, wait, the form is \( y = mx - b' \), but actually the standard form is \( y=mx + b \). If we write it as \( y=mx - (-b) \), but in the given form \( y=\square x-\square \), let's see: \( y=-3x-37 \) can be written as \( y=-3x - 37 \), so the first blank is - 3? No, wait, no. Wait, the slope is - 3, and the y - intercept is - 37? Wait, no. Wait, when \( x = 0 \), \( y=-37 \), so the equation is \( y=-3x-37 \)? Wait, let's re - calculate the slope again. From \( x = 0 \) to \( x = 1 \), \( y \) changes from - 37 to - 40, so the change in \( y \) is \( - 40-(-37)=-3 \), change in \( x \) is \( 1 - 0 = 1 \), so slope \( m=-3 \). The y - intercept \( b \) is the value of \( y \) when \( x = 0 \), which is - 37. So the linear equation is \( y=-3x-37 \)? Wait, but the form is \( y=\square x-\square \). So the first square is - 3? No, wait, no. Wait, maybe I made a mistake. Wait, the form is \( y = ax - b \), and we know that \( y=mx + c \), so if \( y=mx + c=ax - b \), then \( ax - b=mx + c \), so \( a = m \) and \( -b=c \). Since \( c=-37 \), then \( b = 37 \). Wait, let's check: \( y=-3x-(-37) \) is wrong. Wait, no, let's take the standard form \( y=mx + b \), here \( m=-3 \), \( b=-37 \). So \( y=-3x-37 \), which is the same as \( y=-3x-37 \), so in the form \( y=\square x-\square \), the first blank is - 3? No, wait, no. Wait, the slope is - 3, so the coefficient of \( x \) is - 3, and the second blank is 37? Wait, no, let's substitute into the form \( y=\square x-\square \). Let's let the equation be \( y = m x - k \), so \( y=mx - k \). We know that \( y=mx + b \), so \( mx - k=mx + b\implies - k = b\implies k=-b \). Since \( b=-37 \), then \( k = 37 \). And \( m=-3 \). So the equation is \( y=-3x - 37 \)? Wait, no, \( y=-3x-37 \) is equal to \( y=-3x-37 \), so in the form \( y=\square x-\square \), the first box is - 3 and the second box is 37? Wait, no, when \( x = 0 \), \( y=-37 \), so \( y=-3x-37 \) is correct. Let's check the form: \( y=\square x-\square \), so \( y=-3x-37 \) means the first blank is - 3 and the second blank is 37? Wait, no, that can't be. Wait, maybe I messed up the sign. Let's re - express the linear equation. The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-40 + 37}{1-0}=-3 \). The equation is \( y=-3x + b \). When \( x = 0 \), \( y=-37 \), so \( b=-37 \). So \( y=-3x-37 \) (because \( +(-37)=-37 \)). So in the form \( y=\square x-\square \), it's \( y=-3x-37 \), so the first blank is - 3 and the second blank is 37? Wait, no, that would be \( y=-3x - 37 \), which is correct. Let's verify with \( x = 1 \): \( y=-3(1)-37=-3 - 37=-40 \), correct. \( x = 2 \): \( y=-3(2)-37=-6 - 37=-43 \), correct. \( x = 3 \): \( y=-3(3)-37=-9 - 37=-46 \), correct.

Answer:

The first blank is \(-3\) and the second blank is \(37\), so the equation is \(y = - 3x-37\) (filling the blanks as \(-3\) and \(37\) respectively).