To find the equation of the straight line [tex]\( y = mx + c \)[/tex] that fits the given data points, we need to determine the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( c \)[/tex].
1. Identify Two Points From the Table:
We'll use the points [tex]\((-4, 21)\)[/tex] and [tex]\((-3, 18)\)[/tex] to calculate the slope.
2. Calculate the Slope [tex]\( m \)[/tex]:
The formula to calculate the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the points [tex]\((-4, 21)\)[/tex] and [tex]\((-3, 18)\)[/tex]:
[tex]\[
m = \frac{18 - 21}{-3 - (-4)} = \frac{18 - 21}{-3 + 4} = \frac{-3}{1} = -3
\][/tex]
So, the slope [tex]\( m \)[/tex] is [tex]\( -3 \)[/tex].
3. Calculate the y-intercept [tex]\( c \)[/tex]:
To find the y-intercept, we use the equation of the line [tex]\( y = mx + c \)[/tex]. We can substitute one of the points along with the slope and solve for [tex]\( c \)[/tex].
Let's use the point [tex]\((-4, 21)\)[/tex]:
[tex]\[
y = mx + c
\][/tex]
Substituting [tex]\( y = 21 \)[/tex], [tex]\( x = -4 \)[/tex], and [tex]\( m = -3 \)[/tex]:
[tex]\[
21 = (-3)(-4) + c
\][/tex]
Simplifying:
[tex]\[
21 = 12 + c
\][/tex]
Solving for [tex]\( c \)[/tex]:
[tex]\[
c = 21 - 12 = 9
\][/tex]
So, the y-intercept [tex]\( c \)[/tex] is [tex]\( 9 \)[/tex].
4. Formulate the Equation:
Now that we have both [tex]\( m = -3 \)[/tex] and [tex]\( c = 9 \)[/tex], we can write the equation of the line:
[tex]\[
y = -3x + 9
\][/tex]
Therefore, the equation that fits the given data points is:
[tex]\[ y = -3x + 9 \][/tex]
