LINES IN A PLANE

LINES IN A PLANELINEAR EQUATIONa. In one variableA linear equation in one variable (x or y) is an equation of the form:Its graph:The graph of a linear equation in one variable is either:a horizontal line; ora vertical line.examples

Solution:Solve for the value of x.examples

Solution:Solve for the value of y.B. In two variablesA linear equation in two variables is an equation of the form:Where A, B and C are constants and A and B are BOTH nonzero.LINEAR EQUATIONSOME FACTS:It is also known as First-Degree EquationThe highest exponent of BOTH x and y is 1.It doesnt have ANY xy- term.Tell whether the given equation is LINEAR or NOT. Justify your answer.exercisesLINEARLINEARLINEARNOT LinearNOT LinearIts graph:Its graph is the set of all points (a,b) that satisfy the given linear equation in two variables.Its graph is ALWAYS a straight line.examples

Solution:If x = 0, y = 5examples

Solution:If x = 0, y = -3If y = 0, x = 5A point on the lineExamples:Slope of a lineIt is the steepness of a line.It is commonly known as the rise over run.It is the ratio of change in y to change in x.examplesSketch and determine the slope of the line which passes through each pair of points:1. (4, 1), (2,-3)2. (-3, 2), (6,-5)3. (-1, -5), (-3,-5)4. (8, -3), (8,5)141. (4, 1), (2,-3)

Rising line

falling line2. (-3, 2), (6,-5)

horizontal line3. (-1, -5), (-3,-5)

vertical line4. (8, -3), (8,5)Slope and line descriptionSLOPELINE DESCRIPTIONm >0Rising Linem < 0Falling Linem = 0Horizontal/Parallel to x-axism = undefinedVertical/Parallel to y-axisSpecial forms of the equation of a lineUse this when the slope and a point on the line are given.1. POINT-SLOPE FORMFormula:ExampleDetermine the equation of the line passing through point P(2,1) and with a slope of 3.Given: m = 3 and P(2,1)Solution:Use this when two points on the line are given.2. TWO-POINT FORMFormula:ExampleDetermine the equation of the line passing through points (-2,8) and (3,1)Given: (-2,8) and (3,1)Solution:Use this when the slope and the y-intercept (which is equal to b) are given.3. SLOPE-INTERCEPT FORMFormula:ExampleSolution:Use this when the x- and y- intercepts of the line are given.4. TWO-INTERCEPT FORMFormula:Restriction: Cant be used when the line passes through the origin because a=0, b=0.ExampleDetermine the equation of the line whose x-intercept is 2 and y-intercept is -3.Solution:PARALLEL LINESWhen two lines are parallel, it means that their slopes are equal.When the slope of the lines are equal, then they are parallel.Example 1Solution:Transform first the given equation to slope-intercept form.Now, we know the slope and the point on the lineWhat formula to use?POINT-SLOPE FORMsolutionGiven: m = -2 ; P(-2, 3)Example 2Solution:Transform first the given equation to slope-intercept form.Now, we know the slope and the point on the lineWhat formula to use?POINT-SLOPE FORMsolutionGiven: m = 3 ; P(5, -2)Perpendicular LINESWhen two lines are perpendicular to each other, it means that the product of their slopes is -1.If the slope of one line is the negative reciprocal of the slope of the other line, or the product of their slopes is -1, then the lines are PERPENDICULAR.Example 1Solution:Transform first the given equation to slope-intercept form.What formula to use?POINT-SLOPE FORMNEGATIVERECIPROCALsolutionGiven: Example 2Solution:Transform first the given equation to slope-intercept form.What formula to use?POINT-SLOPE FORMNEGATIVERECIPROCALsolutionGiven: m = 2 P(0, -2)