Derive the equation of the line in slope-intercept form.
To derive the equation of a line in slope-intercept form, we typically start with the point-slope form of a line:
\(y - y_1 = m(x - x_1)\)
Where:
- \(m\) is the slope of the line
- \((x_1, y_1)\) is a point on the line
To convert this equation into slope-intercept form (\(y = mx + b\)), we need to isolate the variable \(y\).
Let's go step by step:
1. Start with the point-slope form:
\(y - y_1 = m(x - x_1)\)
2. Expand the terms on the right side:
\(y - y_1 = mx - mx_1\)
3. Add \(y_1\) to both sides to isolate \(y\):
\(y = mx - mx_1 + y_1\)
4. Rearrange the terms in ascending order:
\(y = mx + y_1 - mx_1\)
5. Combine the constants \(y_1 - mx_1\) into a single constant:
\(y = mx + b\)
Where:
- The constant \(b\) is equal to \(y_1 - mx_1\).
Therefore, the equation of the line in slope-intercept form is \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept (the y-coordinate of the point where the line intersects the y-axis).
To derive the equation of a line in slope-intercept form, we need two key pieces of information: the slope of the line (m) and the y-intercept (b).
The slope-intercept form of a linear equation is given by the equation: y = mx + b
The slope (m) represents the rate of change of y with respect to x, and the y-intercept (b) represents the value of y when x is zero.
To find the equation of a line in slope-intercept form, follow these steps:
Step 1: Determine the slope (m) of the line. The slope is given by the formula:
m = (change in y) / (change in x)
Step 2: Identify the coordinates of any point on the line, denoted as (x1, y1).
Step 3: Substitute the values of the slope (m) and the coordinates (x1, y1) into the point-slope formula:
y - y1 = m(x - x1)
Step 4: Simplify and rearrange the point-slope form equation to get the slope-intercept form (y = mx + b) by isolating y:
y - y1 = m(x - x1)
y = m(x - x1) + y1
y = mx - mx1 + y1
The equation in slope-intercept form is:
y = mx - mx1 + y1
With this equation, you now have the equation of the line in slope-intercept form.
To derive the equation of a line in slope-intercept form, you need two key pieces of information: the slope (m) of the line and the y-intercept (b). With these two values, you can write the equation of a line as:
y = mx + b
where y is the dependent variable (usually representing the vertical axis) and x is the independent variable (usually representing the horizontal axis).
The slope (m) represents the rate at which the line is ascending or descending. It can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are any two points on the line.
Once you have the slope (m), you can substitute it into the slope-intercept equation, along with the y-intercept (b), to get the equation of the line.
Let's say you have a line with a slope of 2 and a y-intercept of 3. Plugging these values into the slope-intercept equation, we can write the equation as:
y = 2x + 3
This equation represents a line with a slope of 2 and a y-intercept of 3.