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Inside Collection:

Collection by: Rupinder Sekhon. E-mail the author

# Linear Equations

Module by: Rupinder Sekhon. E-mail the author

Summary: This chapter covers principles of linear equations. After completing this chapter students should be able to: graph a linear equation; find the slope of a line; determine an equation of a line; solve linear systems; and complete application problems using linear equations.

## Chapter Overview

In this chapter, you will learn to:

1. Graph a linear equation.
2. Find the slope of a line.
3. Determine an equation of a line.
4. Solve linear systems.
5. Do application problems using linear equations.

## Graphing a Linear Equation

Equations whose graphs are straight lines are called linear equations. The following are some examples of linear equations:

2x3y=62x3y=6 size 12{2x - 3y=6} {}, 3x=4y73x=4y7 size 12{3x=4y - 7} {}, y=2x5y=2x5 size 12{y=2x - 5} {}, 2y=32y=3 size 12{2y=3} {}, and x2=0x2=0 size 12{x - 2=0} {}.

A line is completely determined by two points, therefore, to graph a linear equation, we need to find the coordinates of two points. This can be accomplished by choosing an arbitrary value for xx size 12{x} {} or yy size 12{y} {} and then solving for the other variable.

### Example 1

#### Problem 1

Graph the line: y=3x+2y=3x+2 size 12{y=3x+2} {}

### Example 2

#### Problem 1

Graph the line: 2x+y=42x+y=4 size 12{2x+y=4} {}

The points at which a line crosses the coordinate axes are called the intercepts. When graphing a line, intercepts are preferred because they are easy to find. In order to find the x-intercept, we let y=0y=0 size 12{y=0} {}, and to find the y-intercept, we let x=0x=0 size 12{x=0} {}.

### Example 3

#### Problem 1

Find the intercepts of the line: 2x3y=62x3y=6 size 12{2x - 3y=6} {}, and graph.

### Example 4

#### Problem 1

Graph the line given by the parametric equations: x=3+2tx=3+2t size 12{x=3+2t} {}, y=1+ty=1+t size 12{y=1+t} {}

### Horizontal and Vertical Lines

When an equation of a line has only one variable, the resulting graph is a horizontal or a vertical line.

The graph of the line x=ax=a size 12{x=a} {}, where aa size 12{a} {} is a constant, is a vertical line that passes through the point ( aa size 12{a} {}, 0). Every point on this line has the x-coordinate aa size 12{a} {}, regardless of the y-coordinate.

The graph of the line y=by=b size 12{y=b} {}, where bb size 12{b} {} is a constant, is a horizontal line that passes through the point (0, bb size 12{b} {}). Every point on this line has the y-coordinate bb size 12{b} {}, regardless of the x-coordinate.

#### Example 5

##### Problem 1

Graph the lines: x=2x=2 size 12{x= - 2} {} , and y=3y=3 size 12{y=3} {}.

## Slope of a Line

### Section Overview

In this section, you will learn to:

1. Find the slope of a line if two points are given.
2. Graph the line if a point and the slope are given.
3. Find the slope of the line that is written in the form y=mx+by=mx+b size 12{y= ital "mx"+b} {}.
4. Find the slope of the line that is written in the form Ax+By=cAx+By=c size 12{ ital "Ax"+ ital "By"=c} {}.

In the last section, we learned to graph a line by choosing two points on the line. A graph of a line can also be determined if one point and the "steepness" of the line is known. The precise number that refers to the steepness or inclination of a line is called the slope of the line.

From previous math courses, many of you remember slope as the "rise over run," or "the vertical change over the horizontal change" and have often seen it expressed as:

riserun, vertical changehorizontal change, ΔyΔxetc.riserun size 12{ { {"rise"} over {"run"} } } {}, vertical changehorizontal change size 12{ { {"vertical change"} over {"horizontal change"} } } {}, ΔyΔx size 12{ { {Δy} over {Δx} } } {}etc.
(5)

We give a precise definition.

Definition 1:

If ( x1x1 size 12{x rSub { size 8{1} } } {}, y1y1 size 12{y rSub { size 8{1} } } {}) and ( x2x2 size 12{x rSub { size 8{2} } } {}, y2y2 size 12{y rSub { size 8{2} } } {}) are two different points on a line, then the slope of the line is

Slope = m = y 2 y 1 x 2 x 1 Slope = m = y 2 y 1 x 2 x 1 size 12{"Slope"=m= { {y rSub { size 8{2} } - y rSub { size 8{1} } } over {x rSub { size 8{2} } - x rSub { size 8{1} } } } } {}

### Example 6

#### Problem 1

Find the slope of the line that passes through the points (-2, 3) and (4, -1), and graph the line.

### Example 7

#### Problem 1

Find the slope of the line that passes through the points (2, 3) and (2, -1), and graph.

### Example 8

#### Problem 1

Graph the line that passes through the point (1, 2) and has slope 3434 size 12{ - { {3} over {4} } } {} .

### Example 9

#### Problem 1

Find the slope of the line 2x+3y=62x+3y=6 size 12{2x+3y=6} {}.

### Example 10

#### Problem 1

Find the slope of the line y=3x+2y=3x+2 size 12{y=3x+2} {}.

### Example 11

#### Problem 1

Determine the slope and y-intercept of the line 2x+3y=62x+3y=6 size 12{2x+3y=6} {}.

## Determining the Equation of a Line

### Section Overview

In this section, you will learn to:

1. Find an equation of a line if a point and the slope are given.
2. Find an equation of a line if two points are given.

So far, we were given an equation of a line and were asked to give information about it. For example, we were asked to find points on it, find its slope and even find intercepts. Now we are going to reverse the process. That is, we will be given either two points, or a point and the slope of a line, and we will be asked to find its equation.

An equation of a line can be written in two forms, the slope-intercept form or the standard form.

The Slope-Intercept Form of a Line: y = mx + b y = mx + b size 12{y= ital "mx"+b} {}

A line is completely determined by two points, or a point and slope. So it makes sense to ask to find the equation of a line if one of these two situations is given.

### Example 12

#### Problem 1

Find an equation of a line whose slope is 5, and y-intercept is 3.

### Example 13

#### Problem 1

Find the equation of the line that passes through the point (2, 7) and has slope 3.

### Example 14

#### Problem 1

Find an equation of the line that passes through the points (–1, 2), and (1, 8).

### Example 15

#### Problem 1

Find an equation of the line that has x-intercept 3, and y-intercept 4.

The Standard form of a Line: Ax + By = C Ax + By = C size 12{ ital "Ax"+ ital "By"=C} {}

Another useful form of the equation of a line is the Standard form.

Let LL size 12{L} {} be a line with slope mm size 12{m} {}, and containing a point (x1,y1)(x1,y1) size 12{ $$x rSub { size 8{1} } ,y rSub { size 8{1} }$$ } {}. If (x,y)(x,y) size 12{ $$x,y$$ } {} is any other point on the line LL size 12{L} {}, then by the definition of a slope, we get

m = y y 1 x x 1 m = y y 1 x x 1 size 12{m= { {y - y rSub { size 8{1} } } over {x - x rSub { size 8{1} } } } } {}
(21)
y y 1 = m ( x x 1 ) y y 1 = m ( x x 1 ) size 12{y - y rSub { size 8{1} } =m $$x - x rSub { size 8{1} }$$ } {}
(22)

The last result is referred to as the point-slope form or point-slope formula. If we simplify this formula, we get the equation of the line in the standard form, Ax+By=CAx+By=C size 12{ ital "Ax"+ ital "By"=C} {}.

### Example 16

#### Problem 1

Using the point-slope formula, find the standard form of an equation of the line that passes through the point (2, 3) and has slope –3/5.

### Example 17

#### Problem 1

Find the standard form of the line that passes through the points (1, -2), and (4, 0).

### Example 18

#### Problem 1

Write the equation y=2/3x+3y=2/3x+3 size 12{y= - 2/3x+3} {} in the standard form.

### Example 19

#### Problem 1

Write the equation 3x4y=103x4y=10 size 12{3x - 4y="10"} {} in the slope-intercept form.

Finally, we learn a very quick and easy way to write an equation of a line in the standard form. But first we must learn to find the slope of a line in the standard form by inspection.

By solving for yy size 12{y} {}, it can easily be shown that the slope of the line Ax+By=CAx+By=C size 12{ ital "Ax"+ ital "By"=C} {} is A/BA/B size 12{ - A/B} {}. The reader should verify.

### Example 20

#### Problem 1

Find the slope of the following lines, by inspection.

1. 3x5y=103x5y=10 size 12{3x - 5y="10"} {}
2. 2x+7y=202x+7y=20 size 12{2x+7y="20"} {}
3. 4x3y=84x3y=8 size 12{4x - 3y=8} {}

Now that we know how to find the slope of a line in the standard form by inspection, our job in finding the equation of a line is going to be very easy.

### Example 21

#### Problem 1

Find an equation of the line that passes through (2, 3) and has slope 4/54/5 size 12{ - 4/5} {}.

If you use this method often enough, you can do these problems very quickly.

## Applications

Now that we have learned to determine equations of lines, we get to apply these ideas in real-life equations.

### Example 22

#### Problem 1

A taxi service charges $0.50 per mile plus a$5 flat fee. What will be the cost of traveling 20 miles? What will be cost of traveling xx size 12{x} {} miles?

### Example 23

#### Problem 1

The variable cost to manufacture a product is $10 and the fixed cost$2500. If xx size 12{x} {} represents the number of items manufactured and yy size 12{y} {} the total cost, write the cost function.

### Example 24

#### Problem 1

It costs $750 to manufacture 25 items, and$1000 to manufacture 50 items. Assuming a linear relationship holds, find the cost equation, and use this function to predict the cost of 100 items.

### Example 25

#### Problem 1

The freezing temperature of water in Celsius is 0 degrees and in Fahrenheit 32 degrees. And the boiling temperatures of water in Celsius, and Fahrenheit are 100 degrees, and 212 degrees, respectively. Write a conversion equation from Celsius to Fahrenheit and use this equation to convert 30 degrees Celsius into Fahrenheit.

### Example 26

#### Problem 1

The population of Canada in the year 1970 was 18 million, and in 1986 it was 26 million. Assuming the population growth is linear, and x represents the year and y the population, write the function that gives a relationship between the time and the population. Use this equation to predict the population of Canada in 2010.

## More Applications

### Section Overview

In this section, you will learn to:

1. Solve a linear system in two variables.
2. Find the equilibrium point when a demand and a supply equation are given.
3. Find the break-even point when the revenue and the cost functions are given.

In this section, we will do application problems that involve the intersection of lines. Therefore, before we proceed any further, we will first learn how to find the intersection of two lines.

### Example 27

#### Problem 1

Find the intersection of the line y=3x1y=3x1 size 12{y=3x - 1} {} and the line y=x+7y=x+7 size 12{y= - x+7} {}.

### Example 28

#### Problem 1

Find the intersection of the lines 2x+y=72x+y=7 size 12{2x+y=7} {} and 3xy=33xy=3 size 12{3x - y=3} {} by the elimination method.

### Example 29

#### Problem 1

Solve the system of equations x+2y=3x+2y=3 size 12{x+2y=3} {} and 2x+3y=42x+3y=4 size 12{2x+3y=4} {} by the elimination method.

### Example 30

#### Problem 1

Solve the system of equations 3x4y=53x4y=5 size 12{3x - 4y=5} {} and 4x5y=64x5y=6 size 12{4x - 5y=6} {}.

### Supply, Demand and the Equilibrium Market Price

In a free market economy the supply curve for a commodity is the number of items of a product that can be made available at different prices, and the demand curve is the number of items the consumer will buy at different prices. As the price of a product increases, its demand decreases and supply increases. On the other hand, as the price decreases the demand increases and supply decreases. The equilibrium price is reached when the demand equals the supply.

### Example 31

#### Problem 1

The supply curve for a product is y=1.5x+10y=1.5x+10 size 12{y=1 "." 5x+"10"} {} and the demand curve for the same product is y=2.5x+34y=2.5x+34 size 12{y= - 2 "." 5x+"34"} {}, where xx size 12{x} {} is the price and y the number of items produced. Find the following.

1. How many items will be supplied at a price of $10? 2. How many items will be demanded at a price of$10?
3. Determine the equilibrium price.
4. How many items will be produced at the equilibrium price?

### Break-Even Point

In a business, the profit is generated by selling products. If a company sells xx size 12{x} {} number of items at a price PP size 12{P} {}, then the revenue RR size 12{R} {} is PP size 12{P} {} times xx size 12{x} {} , i.e., R=PxR=Px size 12{R=P cdot x} {}. The production costs are the sum of the variable costs and the fixed costs, and are often written as C=mx+bC=mx+b size 12{C= ital "mx"+b} {}, where xx size 12{x} {} is the number of items manufactured.

A company makes a profit if the revenue is greater than the cost, and it shows a loss if the cost is greater than the revenue. The point on the graph where the revenue equals the cost is called the Break-even point.

### Example 32

#### Problem 1

If the revenue function of a product is R=5xR=5x size 12{R=5x} {} and the cost function is y=3x+12y=3x+12 size 12{y=3x+"12"} {}, find the following.

1. If 4 items are produced, what will the revenue be?
2. What is the cost of producing 4 items?
3. How many items should be produced to break-even?
4. What will be the revenue and the cost at the break-even point?

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