Algebra 1 - Linear Equations

Introduction

  • Linear equations are equations of degree one. For example, 3x-5y=17 is a linear equation in two variables.
  • The equation of a straight line is also a linear equation.

Slope or Gradient of a line

  • The slope of a line is the ratio of change in the y-coordinates to the change in the x-coordinates.
  • The slope m of a non-vertical line passing through the points x1,y1 and x2,y2 is given by: m=y2-y1x2-x1,x1x2.
  • The formula for slope is also referred to as rise over run.
  • The slope of a line parallel to the x-axis is zero.
  • The slope of the y-axis is not defined.
  • If two lines are parallel, their slopes are equal.
  • If two lines are perpendicular to each other, the product of their slopes is -1.

Forms of Linear Equation

  • The standard form of a line:

    • In the standard form, a linear equation is expressed as: Ax+By=C, where A,B0.
  • The Slope-Intercept form of a line:

    •  The equation of a line with slope m, and making an intercept c on y-axis is given by y=mx+c.
  • The Point Slope form of a line:

    • The equation of a line with slope m, and passing through a point x1,y1 is given by y-y1=mx-x1.
  • The Two Point form of a line:

    • The equation of a line passing through the points x1,y1 and x2,y2 is y-y1=y2-y1x2-x1x-x1.
  • Equation of a line in Intercept form:

    • The equation of a line that makes intercepts of lengths a and b on the x-axis and y-axis respectively is given by xa+yb=1.

Solved Examples

Question 1: Find the slope of a line passing through the points (-3, 5) and (7, 4).

Solution: Here, x1=-3, y1=5, x2=7, and y2=4

Slope, m=y2-y1x2-x1=4-57--3=-17+3=-110

 

Question 2: Find the equation of a line with slope, m=25 and passing through the point -2,1.

Solution: The point-slope form of a line is given by the equation y-y1=mx-x1

Here, x1=-2, y1=1, and m=25.

So, the desired equation will be y-1=25x--2

5y-1=2x+2

5y-5=2x+4

2x-5y+9=0

 

Question 3: Find the equation of a line passing through the points (1, -3) and (2, 5).

Solution: Here, x1=1, y1=-3, x2=2, and y2=5.

slope m=y2-y1x2-x1=5--32-1=5+31=8

The equation of the line will be y-y1=mx-x1

y--3=8x-1

y+3=8x-8

y+3=8x-8

8x-y=11

Cheat Sheet

  • Linear equations are equations in which the highest power (degree) of the variable is 1.
  • The slope m of a non-vertical line passing through the points x1,y1, and x2,y2 is given by m=y2-y1x2-x1,x1x2.
  • Equation of a line in standard form: Ax+By=C, where A and B are non-zero coefficients of x and y, respectively.
  • The Slope-Intercept form of a line: The equation of a line with slope m, and making an intercept c on y-axis is given by y=mx+c.
  • The Point Slope form of a line: The equation of a line with slope m, and passing through a point x1,y1 is given by y-y1=mx-x1.
  • The Two Point form of a line: The equation of a line passing through the points x1,y1 and x2,y2 is y-y1=y2-y1x2-x1x-x1.
  • Equation of a line in Intercept form: The equation of a line that makes intercepts of lengths a and b on the x-axis and y-axis respectively is given by xa+yb=1.
  • Two lines are parallel if and only if their slopes are equal.
  • Two lines are perpendicular if and only if the product of their slopes is -1.

Blunder Areas

  • Remember that the slope of a vertical line is not defined.
  • In the slope-intercept form of a line, y=mx+cc is the y-intercept and must not be confused with the x-intercept.