Overview
If the linear equation for a line has been determined, the slope of that line can be found by solving for any two points on the line. Similarly, the slope-intercept equation can be used to find the y-intercept and graph the line. If any point on a line and the slope are known, the equation for the line can be found.
Finding Slope
In the figure below, the line is shown with a y intercept at (0, 2) and another point at (4, 0). Currently, the equation is in the form x +2y =4. Solving the equation for y, if 2y = -x +4, then y = -1/2 x +2, and the coordinates of the y intercept are (0,2) and the slope is -1/2. (The slope is also shown on the figure below.) Given a line y =mx +3, use coordinates on the graph to solve for m. Notice that the coefficient of x is also 2. Suppose the equation is 2x +3y =7. Then 3y = -2x +7. If x equals -1, then y equals 3, because 3(3) or 9 equals (-2) (-1) +7. If x equals 2, then y equals 1, because 3(1) = -2(2) +7. The change in y is (1-3) or -2 and the change in x is [(2-(-1)] or (2+1) is 3. Therefore, the slope is -2/3. If the equation 3y = -2x +7 is divided by 3, it then becomes y = (-2/3) x + (7/3), so that the coefficient of x is also -2/3.
Slope-Intercept Equation
An equation of the form y = mx +b is called the slope-intercept equation. In the discussion of finding slope from an equation, the coefficient of x, or the value of m, is the same as the slope. What if the value of x is zero? Then the value of mx is also zero, so the value of y is y = 0 +b, or y =b. The coordinates of that point are (0, b). In other words, the value of b is the y-intercept, because when the value of x is 0, the line crosses the y-axis, also known as the y-intercept.
Graphing with the Slope-Intercept Equation
The equation y = 2x +3 is already in slope-intercept form, so we know that the slope m is equal to 2 and that the y intercept is at b or (0, 3). We will just need one more point to draw the line, so we can go up 2 units to y equals 5 and to the right 1 or x equals 1, so that the coordinates of that point are (1, 5). If x equals 1, then y equals 2 +3 equals 5.
Using the Slope-Intercept Equation
Suppose that you have a line with a point (3, 1) that has a slope of 2. The slope-intercept equation y = mx +b can be used to find the equation, because you have a value of x, a value of y, and a value of m. Therefore, 1 equals 2(3) +b. The value of b equals -5, so the slope-intercept equation is y =2x-5.
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