slope-intercept form

(noun)

A linear equation written in the form y=mx+by = mx + by=mx+b.

Related Terms

  • -intercept
  • slope
  • y-intercept
  • zero
  • x-intercept

Examples of slope-intercept form in the following topics:

  • Slope-Intercept Equations

    • One of the most common representations for a line is with the slope-intercept form.
    • Writing an equation in slope-intercept form is valuable since from the form it is easy to identify the slope and yyy-intercept.  
    • Simply substitute the values into the slope-intercept form to obtain:
    • Let's write the equation 3x+2y=−43x+2y=-43x+2y=−4 in slope-intercept form and identify the slope and yyy-intercept.
    • Now that the equation is in slope-intercept form, we see that the slope m=−32m=-\frac{3}{2}m=−​2​​3​​, and the yyy-intercept b=−2b=-2b=−2.
  • Point-Slope Equations

    • Example: Write the equation of a line in point-slope form, given a point (2,1)(2,1)(2,1) and slope −4-4−4, and convert to slope-intercept form
    • To switch this equation into slope-intercept form, solve the equation for yyy:
    • Example: Write the equation of a line in point-slope form, given point (−3,6)(-3,6)(−3,6) and point (1,2)(1,2)(1,2), and convert to slope-intercept form
    • Graph of the line y−1=−4(x−2)y-1=-4(x-2)y−1=−4(x−2), through the point (2,1)(2,1)(2,1) with slope of −4-4−4, as well as the slope-intercept form, y=−4x+9y=-4x+9y=−4x+9.
    • Use point-slope form to find the equation of a line passing through two points and verify that it is equivalent to the slope-intercept form of the equation
  • Solving Systems Graphically

    • Before successfully solving a system graphically, one must understand how to graph equations written in standard form, or Ax+By=CAx+By=CAx+By=C.
    • To do this, you need to convert the equations to slope-intercept form, or y=mx+by=mx+by=mx+b, where m = slope and b = y-intercept.
    • The best way to convert an equation to slope-intercept form is to first isolate the y variable and then divide the right side by B, as shown below.
    • Now −AB\displaystyle -\frac{A}{B}−​B​​A​​ is the slope m, and CB\displaystyle \frac{C}{B}​B​​C​​ is the y-intercept b.
    • Once you have converted the equations into slope-intercept form, you can graph the equations.
  • Parallel and Perpendicular Lines

    • Recall that the slope-intercept form of an equation is: y=mx+by=mx+by=mx+b and the point-slope form of an equation is: y−y1=m(x−x1)y-y_{1}=m(x-x_{1})y−y​1​​=m(x−x​1​​), both contain information about the slope, namely the constant mmm.
    • Example:  Write an equation of the line (in slope-intercept form) that is parallel to the line y=−2x+4y=-2x+4y=−2x+4 and passes through the point (−1,1)(-1,1)(−1,1)
    • Start with the equation for slope-intercept form and then substitute the values for the slope and the point, and solve for bbb: y=mx+by=mx+by=mx+b.  
    • Example:  Write an equation of the line (in slope-intercept form) that is perpendicular to the line y=14x−3y=\frac{1}{4}x-3y=​4​​1​​x−3 and passes through the point (2,4)(2,4)(2,4)
    • Again, start with the slope-intercept form and substitute the values, except the value for the slope will be the negative reciprocal.  
  • Linear Equations in Standard Form

    • A linear equation written in standard form makes it easy to calculate the zero, or xxx-intercept, of the equation.
    • For example, consider an equation in slope-intercept form: y=−12x+5y = -12x +5y=−12x+5.
    • We know that the y-intercept of a linear equation can easily be found by putting the equation in slope-intercept form.
    • However, the zero, or xxx-intercept of a linear equation can easily be found by putting it into standard form.
    • Note that the yyy-intercept and slope can also be calculated using the coefficients and constant of the standard form equation.
  • What is a Linear Function?

    • Linear functions are algebraic equations whose graphs are straight lines with unique values for their slope and y-intercepts.
    • For example, a common equation, y=mx+by=mx+by=mx+b, (namely the slope-intercept form, which we will learn more about later) is a linear function because it meets both criteria with xxx and yyy as variables and mmm and bbb as constants.  
    • Vertical lines have an undefined slope, and cannot be represented in the form y=mx+by=mx+by=mx+b, but instead as an equation of the form x=cx=cx=c for a constant ccc, because the vertical line intersects a value on the xxx-axis, ccc.  
    • Horizontal lines have a slope of zero and is represented by the form, y=by=by=b, where bbb is the yyy-intercept.  
    • The blue line has a positive slope of 12\frac{1}{2}​2​​1​​ and a yyy-intercept of −3-3−3; the red line has a negative slope of −1-1−1 and a yyy-intercept of 555.
  • Slope

    • Slope is often denoted by the letter mmm.
    • Recall the slop-intercept form of a line, y=mx+by = mx + by=mx+b.
    • Putting the equation of a line into this form gives you the slope (mmm) of a line, and its yyy-intercept (bbb).
    • In other words, a line with a slope of −9-9−9 is steeper than a line with a slope of 777.
    • We can see the slope is decreasing, so be sure to look for a negative slope.
  • The Equation of a Line

    • A common form of a linear equation in the two variables xxx and yyy is:
    • Where mmm (slope) and bbb (intercept) designate constants.
    • The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane.
    • In this particular equation, the constant mmm determines the slope or gradient of that line, and the constant term bbb determines the point at which the line crosses the yyy-axis, otherwise known as the yyy-intercept.
    • Three lines — the red and blue lines have the same slope, while the red and green ones have same y-intercept.
  • Slope and Intercept

    • In the regression line equation the constant mmm is the slope of the line and bbb is the yyy-intercept.
    • Regression analysis is the process of building a model of the relationship between variables in the form of mathematical equations.
    • The constant mmm is slope of the line and bbb is the yyy-intercept -- the value where the line cross the yyy axis.
    • An equation where y is the dependent variable, x is the independent variable, m is the slope, and b is the intercept.
  • Slope and Intercept

    • The concepts of slope and intercept are essential to understand in the context of graphing data.
    • The slope or gradient of a line describes its steepness, incline, or grade.
    • A higher slope value indicates a steeper incline.
    • It also acts as a reference point for slopes and some graphs.
    • Functions which are undefined at x=0x=0x=0 have no yyy-intercept.
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