point-slope equation

Examples of point-slope equation in the following topics:

• Point-Slope Equations

  • The point-slope equation is another way to represent a line; only the slope and a single point are needed.
  • The point-slope equation is a way of describing the equation of a line.
  • Given a slope, $m$, and a point $(x_{1}, y_{1})$, the point-slope equation is:
  • Then plug this point into the point-slope equation and solve for $y$ to get:
  • Plug this point and the calculated slope into the point-slope equation to get:

• Parallel and Perpendicular Lines

  • Recall that the slope-intercept form of an equation is: $y=mx+b$ and the point-slope form of an equation is: $y-y_{1}=m(x-x_{1})$, both contain information about the slope, namely the constant $m$.
  • Example:  Write an equation of the line (in slope-intercept form) that is parallel to the line $y=-2x+4$ and passes through the point $(-1,1)$
  • Start with the equation for slope-intercept form and then substitute the values for the slope and the point, and solve for $b$: $y=mx+b$.  
  • Example:  Write an equation of the line (in slope-intercept form) that is perpendicular to the line $y=\frac{1}{4}x-3$ and passes through the point $(2,4)$
  • Therefore, the new equation has a slope of $-4$, through the point $(2,4)$.  

• Direction Fields and Euler's Method

  • Direction fields, also known as slope fields, are graphical representations of the solution to a first order differential equation.
  • The slope field is traditionally defined for differential equations of the following form:
  • Consider the problem of calculating the shape of an unknown curve which starts at a given point and satisfies a given differential equation.
  • Here, a differential equation can be thought of as a formula by which the slope of the tangent line to the curve can be computed at any point on the curve, once the position of that point has been calculated.The idea is that while the curve is initially unknown, its starting point, which we denote by $A_0$, is known (see ).
  • Then, from the differential equation, the slope to the curve at $A_0$ can be computed, and thus, the tangent line.

• Slope

  • Slope describes the direction and steepness of a line, and can be calculated given two points on the line.
  • Putting the equation of a line into this form gives you the slope ($m$) of a line, and its $y$-intercept ($b$).
  • Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between any two distinct points on a line.
  • Starting with the point on the left, $(0, -3)$, sketch a right triangle, going from the first point to the second point, $(5, 1)$.
  • Calculate the slope of a line using "rise over run" and identify the role of slope in a linear equation

• Solving Systems Graphically

  • A simple way to solve a system of equations is to look for the intersecting point or points of the equations.
  • This point is considered to be the solution of the system of equations.
  • To do this, you need to convert the equations to slope-intercept form, or $y=mx+b$, where m = slope and b = y-intercept.
  • Once you have converted the equations into slope-intercept form, you can graph the equations.
  • To determine the solutions of the set of equations, identify the points of intersection between the graphed equations.

• Slope-Intercept Equations

  • If an equation is not in slope-intercept form, solve for $y$ and rewrite the equation.
  • Let's write the equation $3x+2y=-4$ in slope-intercept form and identify the slope and $y$-intercept.
  • Now that the equation is in slope-intercept form, we see that the slope $m=-\frac{3}{2}$, and the $y$-intercept $b=-2$.
  • The value of the slope dictates where to place the next point.
  • Thus we arrive at the point $(2,-5)$ on the line.

• Slope and Y-Intercept of a Linear Equation

  • For the linear equation y = a + bx, b = slope and a = y-intercept.
  • the y coordinate of the point ( 0,a ) where the line crosses the y-axis.
  • A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is y = 25 + 15x.
  • What is the y-intercept and what is the slope?
  • The slope is 15 (b = 15).

• Linear and Quadratic Equations

  • In this particular equation, the constant $m$ determines the slope or gradient of that line, and the constant term $b$ determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.
  • These equations are often referred to as the "equations of the straight line. " In what follows, $x$, $y$, $t$, and $\theta$ are variables, $m$ is the slope, and $b$ is the y-intercept.
  • If $B$ is nonzero, then the y-intercept, or the y-coordinate of the point where the graph crosses the y-axis (where $x$ is zero), is $\displaystyle \frac{C}{B}$, and the slope of the line is $\displaystyle -\frac{A}{B}$.
  • where $(x_1, y_1)$ is any point on the line.
  • The point-slope form expresses the fact that the difference in the y-coordinate between two points on a line (that is, $y-y_1$) is proportional to the difference in the x-coordinate (that is, $x-x_1$).

• Linear Equations in Standard Form

  • For example, consider an equation in slope-intercept form: $y = -12x +5$.
  • We know that the y-intercept of a linear equation can easily be found by putting the equation in slope-intercept form.
  • The zero is the point $(5, 0)$.
  • Note that the $y$-intercept and slope can also be calculated using the coefficients and constant of the standard form equation.
  • If $B$ is non-zero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (where $x$ is zero), is $\frac{C}{B}$, and the slope of the line is $-\frac{A}{B}$.

• Slope and Intercept

  • A higher slope value indicates a steeper incline.
  • Slope is normally described by the ratio of the "rise" divided by the "run" between two points on a line.
  • This is described by the following equation:
  • It also acts as a reference point for slopes and some graphs.
  • As such, these points satisfy $y=0$.