How to Calculate a Slope & Y Intercept

A slope and Y-intercept are characteristics of linear equation plots. Such plots are always straight lines while linear equations are given in the form: Y=aX+b; "a" and "b" are coefficients. Y-intercept is the Y-coordinate of the point where the plot crosses the Y-axis. The slope is the ratio between Y and X-coordinate differences for any two points that belong to the plot. Thus, slope=(Y2-Y1)/(X2-X1), and X1,Y1 and X2,Y2 are coordinates of the first and the second points, respectively. Two such points unambiguously define a linear equation. As an example, calculate the slope and the Y-intercept if the graph passes through two points with coordinates X1=2, Y1=13 and X2=5, Y2=25.

Things You'll Need

• Calculator

Instructions

1. Write the linear equations for the first and the second points.
   Y1=aX1+b first point
   Y2=aX2+b second point
   In our example, they are
   13=2a+b and 25=5a+b, respectively.

2. Subtract the first equation from the second one (Step 1).
   Y2=aX2+b
   Y1=aX1+b
   ----------------
   Y2-Y1=aX2-aX1+b-b. This could be written as Y2-Y1=a(X2-X1).

3. Divide both sides of the equation in Step 2 by "X2-X1" to calculate the slope.
   Slope=a=(Y2-Y1)/(X2-X1). Note that the slope is equal to the coefficient "a."
   In our example, the slope would be
   Slope=a=(25-13)/(5-2)=12/3=4.

4. Write the equation for the Y-intercept point. Such a point has the coordinate "X" equals to 0.
   Y_intercept=0a+b
   Y_intercept=b.

5. Subtract any equation obtained in Step 1 from the equation for Y-intercept point (Step 4)
   Y_intercept=b
   Y1=aX1+b
   ------------------------
   Y_intercept-Y1=-aX1.
   Add "Y1" to both sides of this equation to obtain
   Y_interceipt=Y1-aX1=Y1-slope x X1.
   Thus, the Y-intercept is expressed using the slope (Step 3) and coordinates of any point that belongs to the graph.
   In our example, Y-intercept=13-(4 x 2)=13-8=5.

References

• "Algebra 1: Practice Workbook With Examples", Houghton Mifflin School, Jun 2001.