Equation of a straight line in slope-intercept form :
y = mx + b
Let us see see, how to derive the slope-intercept form equation of a straight through the following steps.
Step 1 :
Let L be a line with slope m and y-intercept b. Circle the point that must be on the line. Justify your choice.
(b, 0) (0, b) (0, m) (m, 0)
The coordinate of x is 0 in the point that includes the y-intercept.
Step 2 :
Recall that slope is the ratio of change in y to change in x. Complete the equation for the slope m of the line using the y-intercept (0, b), and another point (x, y) on the line.
Slope m = change in y-values/change in x-values
Slope m = (y - b)/(x - 0)
Slope m = (y - b)/x
Step 3 :
In an equation of a line, we often want y by itself on one side of the equation. Solve the equation from Step 2 for y.
m = (y - b)/x
Multiply both sides by x
mx = [(y - b)/x]x
mx = y - b
Add b to both sides of the equation.
mx + b = (y - b) + b
mx + b = y
Write the equation with y on the left side.
y = mx + b
Critical thinking : Write the equation of a line with slope m that passes through the origin. Explain your reasoning.
y = mx
Because the origin is on the y-axis, the graph crosses the y-axis at (0, 0). So, the y-intercept b is 0, and y = mx + b becomes y = mx.
Problem 1 :
A line is passing through the points (2, 3) and (0, 4). Find the equation of the line in slope intercept form.
Solution :
Step 1 :
Fine the slope of the line using the points (2, 3) and (0, 4).
Slope m = change in y-values/change in x-values
Slope m = (4 - 3)/(0 - 3)
Slope m = 1/(-3)
Slope m = -1/3
Step 2 :
In the point (0, 4), x-coordinate is zero. So, the line intersects y-axis at this point.
Since the y-coordinate at this point is 4, y-intercept is 4.
So, the equation of the line in slope-intercept form is
y = (-1/3)x + 4
Problem 2 :
A line is passing through the points (0, 0) and (-1, -8). Find the equation of the line in slope intercept form.
Solution :
Step 1 :
Fine the slope of the line using the points (0, 0) and (-1, -8).
Slope m = change in y-values/change in x-values
Slope m = (-8 - 0)/(-1 - 0)
Slope m = -8/(-1)
Slope m = 8
Step 2 :
Since the line is passing through the origin (0,0), there is no y-intercept or y-intercept = 0.
So, the equation of the line in slope-intercept form is
y = 8x
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