Sure, let's write a linear equation for the line that passes through the points [tex]\((0, 4)\)[/tex] and [tex]\((2, 1)\)[/tex].
A linear equation is generally of the form [tex]\( y = mx + b \)[/tex], where:
- [tex]\( m \)[/tex] is the slope of the line
- [tex]\( b \)[/tex] is the y-intercept, which is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]
First, we need to determine the slope [tex]\( m \)[/tex] of the line passing through the two given points. The formula for 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]
Given points:
[tex]\( (x_1, y_1) = (0, 4) \)[/tex]
[tex]\( (x_2, y_2) = (2, 1) \)[/tex]
Substituting these values into the slope formula:
[tex]\[ m = \frac{1 - 4}{2 - 0} = \frac{-3}{2} = -1.5 \][/tex]
So, the slope [tex]\( m \)[/tex] of the line is [tex]\( -1.5 \)[/tex].
Next, we need to find the y-intercept [tex]\( b \)[/tex]. We can use the formula for the y-intercept, which is derived from the line equation [tex]\( y = mx + b \)[/tex]. Using the point [tex]\((0, 4)\)[/tex], where [tex]\( x = 0 \)[/tex] and [tex]\( y = 4 \)[/tex]:
Substitute the values [tex]\( x = 0 \)[/tex] and [tex]\( y = 4 \)[/tex] into the equation:
[tex]\[ 4 = -1.5(0) + b \][/tex]
[tex]\[ 4 = 0 + b \][/tex]
[tex]\[ b = 4 \][/tex]
So the y-intercept [tex]\( b \)[/tex] is [tex]\( 4 \)[/tex].
Therefore, the linear equation of the line passing through the points [tex]\((0, 4)\)[/tex] and (2, 1)\) is:
[tex]\[ y = -1.5x + 4 \][/tex]
