Sure! Let's find the slope and y-intercept of the line passing through the points [tex]\( (12, 11) \)[/tex] and [tex]\( (-2, 3) \)[/tex].
Step 1: Calculate the Slope
The formula to determine 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 the points [tex]\((12, 11)\)[/tex] and [tex]\((-2, 3)\)[/tex]:
- [tex]\((x_1, y_1) = (12, 11)\)[/tex]
- [tex]\((x_2, y_2) = (-2, 3)\)[/tex]
Applying these coordinates to the formula:
[tex]\[ m = \frac{3 - 11}{-2 - 12} = \frac{-8}{-14} = \frac{4}{7} \][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(\frac{4}{7}\)[/tex] or approximately 0.5714.
Step 2: Calculate the Y-intercept
The equation of the line is written in the slope-intercept form:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. To find [tex]\( b \)[/tex], we can use one of the given points and the slope we just calculated. Let's use the point [tex]\( (12, 11) \)[/tex]:
Substitute [tex]\( x = 12 \)[/tex], [tex]\( y = 11 \)[/tex], and [tex]\( m = \frac{4}{7} \)[/tex] into the equation:
[tex]\[ 11 = \frac{4}{7} \cdot 12 + b \][/tex]
First, calculate [tex]\( \frac{4}{7} \cdot 12 \)[/tex]:
[tex]\[ \frac{4}{7} \cdot 12 = \frac{48}{7} \approx 6.8571 \][/tex]
Now we have:
[tex]\[ 11 = 6.8571 + b \][/tex]
Solve for [tex]\( b \)[/tex] by subtracting 6.8571 from both sides:
[tex]\[ b = 11 - 6.8571 = 4.1429 \][/tex]
So, the y-intercept [tex]\( b \)[/tex] is approximately 4.1429.
Conclusion:
The slope of the line passing through the points [tex]\( (12, 11) \)[/tex] and [tex]\( (-2, 3) \)[/tex] is approximately 0.5714, and the y-intercept is approximately 4.1429.
Thus, the equation of the line in slope-intercept form is:
[tex]\[ y = 0.5714x + 4.1429 \][/tex]
