Unit 5 Writing the Equation of a Line HSA ALGEBRA Unit 5 Writing the Equation of a Line

You can also find the slope (m) of the line The Equation of a Line Given any two points in the coordinate plane, you can draw a line through them. x y l (X2,Y2) (X1,Y1) rise You can also find the slope (m) of the line run m = Y2 – Y1 X2 – X1 m = rise run

The Equation of a Line y (X2,Y2) x (X1,Y1) A line contains infinite points. x y The Equation of the Line is the relationship between the coordinates of the infinite points on the line. It has three different forms: Slope-Intercept Point-Slope Standard (X2,Y2) (X1,Y1)

The Equation of a Line has three Different Forms Slope-Intercept Y = mX + b where m is the slope and b is the y-intercept Point-Slope Y – Y1 = m (X – X1) where m is the slope and X1 and Y1 are the coordinates of one of the given points Standard AX + BY = C where A, B, and C are integers, and A is positive

The Equation of a Line has three Different Forms We will focus on two Forms: Slope-Intercept Form Y = mX + b, where m is the slope and b is the y-intercept Point-Slope Form Y – Y1 = m (X – X1), where m is the slope and X1 and Y1 are the coordinates of one of the given points

How can you find the Equation of a line? x y l You need two points. They can be the intercepts, y-intercept (0,Y) (X2,Y2) (X1,Y1) They can be any two points, or… …the combination of an intercept and a point. x-intercept (X,0)

Finding the Equation of a Line given two points x y l First step - find the slope (m) of the line (4,6) m = rise run rise m = Y2 – Y1 X2 – X1 (-3,-2) m = 6 – (-2) 4 – (-3) run m = 8 7

Finding the Equation of a Line given two points x y l Second step - substitute the slope (m) of the line and the coordinates of either point in the Point-Slope Form of the Equation (4,6) (-3,-2) rise Y – Y1 = m(X – X1) Y – 6 = 8 (X – 4) 7 run m = 8 7

Finding the Equation of a Line given two points Third step - solve the equation using cross-multiplication, distributive property and opposite operations to isolate the variables to the left and constant (numbers) to the right Y – 6 = 8(X – 4) 7 cross-multiplication 7(Y – 6) = 8(X – 4) distributive property 7Y – 42 = 8X – 32 opposite operations – 8X + 7Y = 42 - 32

Finding the Equation of a Line – Practice 1 Write the equation of the line through points (9, 5) and (-3, -1). First step - find the slope (m) of the line m = Y2 – Y1 X2 – X1 m = – 1 – (5) – 3 – 9 m = – 6 – 12 m = 1 2 Second step - substitute the slope (m) and the coordinates of either point into the Point-Slope Form Y – Y1 = m(X – X1) Y – (– 1) = 1(X – (-3)) 2

Finding the Equation of a Line – Practice 1 Third step - solve the equation using cross-multiplication, distributive property, and opposite operations to isolate the variables and constant. Y – (-1) = 1(X – (-3)) 2 Multiply both sides by two 2(Y + 1) = 1(X + 3) distributive property 2Y + 2 = X + 3 opposite operations 2Y = X + 3 - 2 2Y = X + 1 Solve for Y Y = ½ X – ½

Finding the Equation of a Line – Practice 2 Write the equation of the line with slope –3/4 going through point (-2, 5). First step - find the slope (m) - given m = -3 4 Second step - substitute the slope (m) of the line and the coordinates of the point in the Point-Slope Form Y – Y1 = m(X – X1) Y – 5 = - 3 (X – (-2)) 4

Finding the Equation of a Line – Practice 2 Third step - solve the equation using cross-multiplication, distributive property and opposite operations Y – 5 = - 3(X – (-2)) 4 cross-multiplication 4(Y – 5) = – 3(X + 2) distributive property 4Y – 20 = – 3X - 6 opposite operations 4Y = – 3X - 6 + 20 4Y = – 3X + 14 Y = – ¾ X + 7/2

Finding the Equation of a Line – Practice 3 What is the equation for the line below? x y First step - Find the slope (m) In this case, identify two points on the line. (- 3, 3) y-intercept = -2 m = - 2 – 3 0 – (- 3) m = - 5 3

Finding the Equation of a Line – Practice 3 Second step - substitute the slope (m) of the line and the coordinates of one of the points in the Point-Slope Form of the Equation m = - 5 3 (3, - 3) or (0, - 2) Y – Y1 = m (X – X1) Y – (- 2 ) = - 5 (X – 0) 3

Finding the Equation of a Line – Practice 3 Third step - solve the equation using cross-multiplication, distributive property and opposite operations Y – (- 2) = - 5 (X – 0) 3 cross-multiplication 3(Y + 2) = – 5(X) distributive property 3Y + 6 = – 5X opposite operations 3Y = – 5X – 6 Y = – 5/3 X – 2

Find the Equation of a Line – Practice 4 x y What is the Standard form of the equation for line s? 5x + 2y = 20 What is the slope-intercept form of the equation for line s? y = - 5x + 5 2

Find the Equation of a Line – Practice 4 x y What is the slope for line s? m = - 5/2 What is the x-intercept for line s? x-intercept is 2 (2,0)

Questions ? Good luck!