Unit 2.2

Linear Representation and Equations of Lines

Linear Representations

Linear Data can be represented in a variety of ways:• x-y tables• Graphs• Equations• Verbal representation

Slope-Intercept Form of a Line

y = mx + b

Slope of line Y-intercept of line

x and y are points (x,y) on the line

Practice Problems

Identify the slope (m) and y-intercept (b) of the following lines:

1.y = ½ x – 4 2.y = -3x + 53.y = 6 – 2x4.y = -2/3x + 105. y = -8 + .5x

Writing Linear Equation from Slope and y-intercept

• If we know the slope (m) and y-intercept (b), we can write the equation of a line.

Examples: m = ¾ , b=-6 Equation: y = ¾ x – 6 m = -2, b = 3 Equation: y = -2x + 3 m = 0, b = 5 Equation: y = (0)x + 5 or y = 5

Practice Problems

• Write the equations for the following lines:

1. m = 3, b = -42. m = -1/2, b=23. m = 6, b = 04. m = 2/5, b = -25. m = 9/5, b = 32

Write the equation of the line in slope-intercept form for the following graph:

m = b = Equation:

Writing Equation from the slope and one point

If we know the slope of a line and one point on the line, we can find the y-intercept by doing the following:

1. Replace m with the slope given in the problem2. Use the point we know to replace x and y3. Solve for b

Example 1:

• Find the equation of the line with a slope of 2 that passes through point (1, 6).

• m = 2, x = 1, y = 6

Use the general equation: y = mx + b to find b 6 = 2(1) + b 6 = 2 + b 4 = b Equation of line: y = 2x + 4

Example 2:

• Find the equation of the line with a slope of ½ that passes through (4, 8).

m = ½ , x = 4, y = 8y = mx + b8 = ½(4) + b8 = 2 + b6 = b Equation of line: y = ½ x + 6

Practice Problems

• Write the equation of the line with the following information:

1. m = -3, contains (1, -12)2. m = 4, contains (-1, 6)3. m = -1, contains (0, 3)4. m = 2/3, contains (6, 9)

Writing Equation from Two Points• We can also write the equation if we know

two points on the line:1. Find the slope of the line passing through the

two points (this is m)2. Find the y-intercept by solving for b using the

slope and one point on the line3. Put m and b into the general equation y = mx + b

Example:• Find the equation of the line passing through (1,4)

and (2,2).1. Find slope: m = = -2

2. Find b: y = mx + b m = -2, x = 1, y = 4 y = mx + b 4 = -2(1) + b 4 = -2 + b 6 = b 3. Equation: y = -2x + 6

12

42

Practice Problem

• Find the equation of the line passing through (-2,5) and ( 1,2)

m =

b =

Equation:

Example 1: Joe puts $50 in a savings account and saves $15 a week.

Table Equation GraphWeek Money

Example 2: Cost of fixing a furnace Table: Equation: Graph:Hours Cost 0 $50 1 $75 2 $100 3 $125 4 $150

Verbal:

Example 3: Table EquationX | Y

Standard Form of a Line

• The standard form of a line is ax + by = cwhere a, b, are integers, c is real number and x and y are points on the line

• The standard form of a line is often used to find x- and y-intercepts.