Equations of Lines & Applications of Linear Equations in 2 Variables

Math 021

The following forms of equations can be used to find the equations of lines given various information:

Standard Form: Ax + By = C where A, B, and C are real numbers

Slope-Intercept Form: y = mx + b where m is the slope and the y-intercept is (0,b)

Point-Slope Form: y – y1 = m(x – x1) where m is the slope and the point

(x1, y1) is a point on the line.

Examples – Find the equation of each line. Write the answer in standard form:

a. Slope = 3, y-intercept is (0, -2)

b. Slope = - , y-intercept is

Examples – Find the equation of each line. Write the answer in slope-intercept form:

a. Slope = 4, point on line is (1, 3)

b. Slope = -5, point on line is (-6,-1) c. Slope = , point on line is (4, -3)

d. Slope = - , point on line is (-9,4) 

Examples – Find the equation of each line. Write the answer in standard form:

a. Passing through (8, 6) and (9,9)

b. Passing through (-3,4) and (-1,10)

c. Passing through (-8,-6) and (-6,-7) 

Examples – Find the equation of each line. Write the answer in standard form: a. Parallel to y = -3x + 1 passing through

(2, 5) b. Parallel to 8y – 4x = 2 passing through

(4, -3) c. Perpendicular to passing through (-1, 2)

d. Perpendicular to 6y – 3x = -2 passing through (6, -5)

Applications of Linear Equations in 2 Variables

One of applications of linear equations is using the slope of a line to model real world scenarios. The slope of a line is used to represent various rates of change such as:

- Velocity (i.e. miles per hour, meters per

second) - Cost per unit - Population growth per year

A barrel rolls down a 400 foot incline. After 2 seconds the barrel is rolling 6 ft. per second. After 3 seconds the barrel is rolling at 9 ft. per second. Let t represent the time in seconds, s represent the speed. Assume that the speed of the barrel increases at a linear rate. Use the above information to answer the following:

a. Give two ordered pairs in the form (time, speed) b. Find an equation in slope-intercept form that

represents the relationship between time and speed

c. Use the equation to find the velocity after 5.5 seconds

In 1990, the population of a country was 45,000. In the year 2005, the population of the some country was 60,105. Let x represent years past 1990 and y represent the population. Assume that the population of the country grows at a linear rate. Use the above information to answer the following:

Give two ordered pairs in the form (years past 1990, population)

Find an equation in slope-intercept form that represents the relationship between years after 1990 and population

Use the equation to find the population in the year 1997

A toy company developed a new action figure. After 4 months of being available for purchase, the company sells 52 thousand figures. After 7 months, the company sells 91 thousand figures. Let x represent the number of months of the action figures being available for purchase, let y represent the number of thousands of action figures sold. Assume that the relationship follows a linear pattern.

Give two ordered pairs in the form (months available, thousands of figures sold)

Find an equation in slope-intercept form that represents the relationship between months available for purchase and number of thousands of figures sold

Use the equation to find the number of figures sold 11 months after the action figures are available for purchase.