Linear Equations in Linear Equations in Two VariablesTwo Variables

Linear Equations in Linear Equations in Two VariablesTwo Variables

Graphing Linear EquationsGraphing Linear Equations

Writing Equations and Graphing

• These activities introduce rates of change and defines slope of a line as the ratio of the

vertical change to the horizontal change.

• This leads to graphing a linear equation and writing the equation of a line in three

different forms.

The Coordinate PlaneThe Coordinate PlaneThe Coordinate PlaneThe Coordinate Plane

Graphing ordered pairs in the Graphing ordered pairs in the coordinate planecoordinate plane

The coordinate plane is formed by placing two number lines called coordinate axes so that one is horizontal (x-axis) and one is vertical (y-

axis). These axes intersect at a point called the origin (0, 0),

which is labeled 0.An ordered pair (x, y) is a pair of real numbers that correspond to a point in the coordinate plane. The first number in an ordered pair is the x-coordinate and the second number is the y-

coordinate.

Coordinate axes

originquadran

ts

ordered pair

In a previous lesson, you learned to plot points

from a table of values. The solution to an equation in two

variables is all ordered pairs of real numbers (x,

y) that satisfy the equation.

You can graph the solution to an equation in

two variables on the coordinate plane by using the (x, y) values from the

table.Graph each ordered pair in the table. Do all the points appear

to lie on the line?

Make a table of ordered pairs for each equation. Then graph the

ordered pairs. (x = -3, -2, -1, 0, 1, 2, 3)

1) y = x + 32) y = x – 23) y = 2x4) y = -2x5) y = - 2

6) y = -x + 17) y = x2 – 1

8) y = 39) y = 2x2 - 1

In everyday life, one quantity often depends on

another. On an automobile trip, the

amount of time that you spend driving depends on your speed. In cases like these, it is said that the

second quantity is a function of the first. The second quantity is called

dependent, while the first is independent.

Can you think of other situations in real-life where one quantity is dependent

on another.

Many functions (equation) can be represented by an

equation in two variables. The variable

that represents the dependent quantity is called the dependent variable. The variable

that represents the independent quantity is called the independent

variable. When you write an ordered pair

using these variables (x, y), the independent

variable appears first.(independent, dependent)

In the function y = 2x, each x-value is paired with exactly one y-value. Thus the value of y depends

on the value of x.In a function, the variable of the domain is called the independent variable and the variable of the

range is called the dependent variable. On a graph, the independent variable is represented on the horizontal axis and the

dependent variable is represented on the

vertical axis.

Example

The value v (in cents) of n nickels is given by the

function (equation) v = 5n

a) Identify the independent and

dependent variable.

b) Represent this function in a table for n

= 0, 1, 2, 3, 4, 5, 6.

c) Represent this function in a graph.

Use both a table and a graph to represent each function for the

given values of the independent variable.

1) The value v of 0, 1, 2, 3, 4, 5, and 6 dimes is a function of the

number of dimes, d.

2) The distance d traveled by a jogger who jogs 0.1 miles per

minute over 0, 1, 2, 3, 4, and 5 minutes is a function of time, t.

3) One hat cost $24. The total cost c of 1, 2, 3, 4, 5, and 6 hats is a function of the number n of hats

purchased.

Domain

Range

Try another

What is the independent variable?What is the dependent variable?

What function represents this situation?

A 1200-gallon tank is empty. A valve is opened and water flows into the

tank at the constant rate of 25 gallons per minute.

1) How would you find the amount of water in the tank after 5 minutes, 6

minutes, and 7 minutes?

2) How would you find the time it takes for the tank to contain 500 gallons, 600

gallons, and 700 gallons?

Slope and Rate of Slope and Rate of ChangeChange

Slope and Rate of Slope and Rate of ChangeChange

Finding the slope of a LineFinding the slope of a Line

Slope – A measure of the steepness of a line on a graph; rise divided by the run.If P (x1, y1) and Q (x2, y2) lie along a nonvertical line in the coordinate plane, the line has slope m, given by

Linear equation – An equation whose solutions form a straight line on a coordinate plane.Collinear – Points that lie on the same line.

Remember, linear equations have constant slope. For a line on the coordinate plane, slope is the following ratio. This ratio is

often referred to as “rise over run”.

Slope and Orientation of a

LineIn the coordinate plane, a line with

positive slope rises from left to right. A line with negative

slope falls from left to right. A line with

0 slope is horizontal. The slope of a vertical line is undefined.

Find the slope of the line that passes through each pair of points.

1) (1, 3) and (2, 4)

2) (0, 0) and (6, -3)

3) (2, -5) and (1, -2)

4) (3, 1) and (0, 3)

5) (-2, -8) and (1, 4)

Finding points on a LineGraphing a line using a point and the slope is

shown here.

Graph the line passing through (1, 3) with slope

2.

a) Graph point (1, 3).

b) Because the slope is 2, you can find a second point on the line by

counting up 2 units and then right 1 unit.

C) Draw a line through point (1, 3) and the new

point.

A line with slope –¼ contains P(0, 5).

a) Sketch the line

b) Find the coordinates of a second point on

the line.

Sketch each line from the information given. Find a second point on the line.

a) point H(0, 0), slope 2b) point S(3, 3), slope -3

c) point J( 2, 5); slope –1/2 d) point T(-3, 1); slope – 2/5

Finding Slope from a Graph

Choose two points on the line (-4, 4) and (8, -2). Count the rise over run or you can use the slope formula. Notice if you switch (x1, y1) and

(x2, y2), you get the same slope:

Real-world ApplicationsYou have been solving

problems in which quantities change since early school

days. For instance, here is a typical first-grade problem:

If I have two apples, and then I buy five more

apples, how many apples do I have in all?

This situation concerns a change in the number of

apples.

Try this problem

150/3 compares quantities

and is called rate.

One marker on a road showed 20 miles. The next marker

showed 170 miles. How many miles have gone by?

The time was 2:00 pm at the first marker. Then it was 5:00

pm at the second marker. How may hours have passed?

As you can see, simple subtraction is all that is required to answer the

questions.

The answers, 150 miles and 3 hours, indicate a change in miles and a change in time,

respectively.

Since each quantity describes a change, the expression is a rate of change. You read the rate

as150 miles in 3 hours

or150 miles per 3 hours.

In everyday life, most rates are given “per one unit” of a

quantity. This is called a unit rate. So, it is more common to perform the division 150/3 = 50

to arrive at a unit rate of change:

50 miles per 1 houror

50 miles per hour.

Rates can be positive (+) or

negative (-)

After stopping to buy gas, a motorist drives at

a constant rate as indicated below.

Time (t) 0 hrs 2 hrs

Dist (d) 5 mi120 mi

Find the speed or rate of change for the car then

graph the situation.

Problem solving

Calculate each rate of change

1) A motorist drives 140 miles in 3 hours and 280

miles in 4.4 hours.

2) After 5 minutes a temperature of 35° F was

recorded. After 6.5 minutes, the

temperature was 32° F.

3) A hose flows 250 gallons of water in 2

minutes and 875 gallons in 7 minutes.

4) A small aircraft begins a descent to land. At 1.5 minutes the altitude is 5450 feet, and at 3 minutes the altitude is 4700 feet. If the plane descends at

a constant rate, what is the rate of descent.

5) 32 people exit the stadium in 2.5 minutes and 384 people leave the stadium in 30 minutes.

6) A race car drives 2 miles in 0.02 hours and 305 miles in 3.02 hours.

7) After 3.5 minutes of descent, a small plane’s altitude is 3425 feet. After 6.0 minutes, the

plane’s altitude if 2375.8) Does the following data indicate a constant

speed? Explain.2 mi/4 min 8 mi/16 min 10 mi/24 min

A linear equation is an equation whose

solutions fall on a line on the

coordinate plane. All solutions of a

particular linear equation fall on the

line, and all the points on the line are

solutions of the equation.

Look at the graph to the left, points (1, 3)

and (-3, -5) are found on the line

and are solutions to the equation.

If an equation is linear, a constant change in

the x-value produces a constant change in the

y-value.

The graph to the right shows an example

where each time the x-value increases by

2, the y-value increases by

3.

The equationy = 2x + 6

is a linear equation because it is the

graph of a straight line and each time x increases by 1

unit, y increases by 2

X Y = 2x + 6 Y (x, y)

1 2(1) + 6 8 (1, 8) 2 2(2) + 6 10 (2, 10) 3 2(3) + 6 12 (3, 12) 4 2(4) + 6 14 (4, 14) 5 2(5) + 6 16 (5, 16)

Using Slopes and Using Slopes and InterceptsIntercepts

Using Slopes and Using Slopes and InterceptsIntercepts

x-intercepts and y-interceptsx-intercepts and y-intercepts

x-intercept – the x-coordinate of the point where the graph of a line crosses the x-axis (where y = 0).y-intercept – the y-coordinate of the point where the graph of a line crosses the y-axis (where x = 0).Slope-intercept form (of an equation) – a linear equation written in the form y = mx +b, where m represents slope and b represents the y-intercept.Standard form (of an equation) – an equation written in the form of Ax + By = C, where A, B, and C are real numbers, and A and B are both ≠ 0.

Standard Form of an Equation• The standard form of

a linear equation, you can use the x- and y- intercepts to make a graph.

• The x-intercept is the x-value of the point where the line crosses.

• The y-intercept is the y-value of the point where the line crosses.

Ax + By = C

To graph a linear equation in standard form, you fine the x-intercept by

substituting 0 for y and solving for x. Then substitute 0 for x and solve for y.

2x + 3y = 62x + 3(0) = 6 2x = 6 x = 3

The x-intercept is 3.(y = 0)

2x + 3y = 62(0) + 3y = 6 3y = 6 y = 2

The y-intercept is 2. (x = 0)

Find the x-intercept and y-intercept of each line. Use the intercepts to graph the

equation.

1) x – y = 5

2) 2x + 3y = 12

3) 4x = 12 + 3y

4) 2x + y = 7

5) 2y = 20 – 4x

Slope-intercept FormSlope-intercept FormSlope-intercept FormSlope-intercept Form

y = mx + by = mx + b

Slope-intercept Form

• An equation whose graph is a straight line is a linear equation. Since a function rule is an equation, a function can also be linear.

• m = slope• b = y-intercept

Y = mx + b(if you know the slope and

where the line crosses the y-axis, use this form)

For example in the equation;y = 3x + 6

m = 3, so the slope is 3b = +6, so the y-intercept is +6

Let’s look at another:y = 4/5x -7

m = 4/5, so the slope is 4/5b = -7, so the y-intercept is -7

Please note that in the slope-intercept formula;y = mx + b

the “y” term is all by itself on the left side of the equation.

That is very important!

WHY?If the “y” is not all by itself, then we must first use the rules of algebra to isolate the “y” term.

For example in the equation:

2y = 8x + 10

You will notice that in order to get “y” all by itself we have to divide both sides by 2.

After you have done that, the equation becomes:Y = 4x + 5

Only then can we determine the slope (4), and the y-intercept (+5)

OK…getting back to the lesson…Your job is to write the equation of a line

after you are given the slope and y-intercept…

Let’s try one…

Given “m” (the slope remember!) = 2And “b” (the y-intercept) = +9

All you have to do is plug those values intoy = mx + b

The equation becomes…y = 2x + 9

Let’s do a couple more to make sure you are expert at this.

Given m = 2/3, b = -12,Write the equation of a line in slope-intercept form.

Y = mx + bY = 2/3x – 12

One last example…Given m = -5, b = -1

Write the equation of a line in slope-intercept form.Y = mx + bY = -5x - 1

Writing an Equation From a Graph

You can write an equation from a graph. Use 2

points to find the slope. Then use the slope and the y-intercept to write

the equation.Step 1 Using the slope

formula, find the slope. Two points on the line are (0, 2) and

(4, -1).Step 2 write an equation in

slope-intercept form. The y-intercept is 2.

Write an equation of each line.

Use points (0, 1) and (-2, 0)

Use points (0, 1) and (3, -1)

Given the slope and y-intercept, write the equation of a line in slope-intercept form.

1) m = 3, b = -14

2) m = -½, b = 4

3) m = -3, b = -7

4) m = 1/2 , b = 0

5) m = 2, b = 4

Using slope-intercept form to find slopes and

y-intercepts

The graph at the right shows the equation of a

line both in standard form and slope-intercept

form.

You must rewrite the equation 6x – 3y = 12 in slope-intercept to be able to identify the slope and

y-intercept.

Using slope-intercept form to write equations, Rewrite the equation solving

for y = to determine the slope and y-intercept.

3x – y = 14-y = -3x + 14-1 -1 -1y = 3x – 14 or 3x – y = 14 3x = y + 143x – 14 = y

x + 2y = 82y = -x + 8 2 2 2y = -1x + 4 2

Write each equation in slope-intercept form.

Identify the slope and y-intercept.

2x + y = 10

-4x + y = 6

4x + 3y = 9

2x + y = 3

5y = 3x

Write the equation of a line in slope-intercept form that passes through points

(3, -4) and (-1, 4).

1) Find the slope.

4 – (-4) 8-1 – 3 -4m = -2

2) Choose either point and substitute. Solve for b.

y = mx + b (3, -4)-4 = (-2)(3) + b-4 = -6 + b2 = bSubstitute m and b in

equation.Y = mx + bY = -2x + 2

Write the equation of the line in slope-intercept form that passes through

each pair of points.

1) (-1, -6) and (2, 6)

2) (0, 5) and (3, 1)

3) (3, 5) and (6, 6)

4) (0, -7) and (4, 25)

5) (-1, 1) and (3, -3)

Graphing an Equationy = 3x -1

The y-intercept is -1, so plot point (0, -1)

The slope is 3, use the slope to plot the second point

Draw a line through the two points.

Point-Slope FormPoint-Slope FormPoint-Slope FormPoint-Slope Form

Writing an equation when you Writing an equation when you know a point (2, 5) and the know a point (2, 5) and the

slope m = 2slope m = 2

In the graph below, use the information provided to write the equation of the

line. Use what you know about writing an equation in slope-intercept form.

Slope = 2 and point (2,7)

Do you think you can use the same method to find the y-intercept in the graph below?

Here we must use a different form of writing an equation and that form is called point-

slope.

Slope = 7/3 and point (2,7)

Point-Slope Form and Writing

Equations

• Suppose you know that a line passes through the point (3, 4) with slope 2. You can quickly write an equation of the line using the x- and y-coordinates of the point and using the slope.

• The point-slope form of the equation of a nonvertical line that passes through the

(x1, y1) with slope m.

y – y1 = m(x – x1)

(if you know a point and the slope, use this form)

Let’s try a couple.

Using point-slope form, write the equation of a line that passes through (4, 1) with slope -2.

y – y1 = m(x – x1)

y – 1 = -2(x – 4)Substitute 4 for x1, 1 for y1 and -2 for m.

Write in slope-intercept form.y – 1 = -2x + 8

y = -2x + 9

One last example

Using point-slope form, write the equation of a line that passes through (-1, 3) with slope 7.

y – y1 = m(x – x1)

y – 3 = 7[x – (-1)]y – 3 = 7(x + 1)

Write in slope-intercept formy – 3 = 7x + 7y = 7x + 10

If you know two points on a line, first use them to find the slope. Then you can write

an equation using either point.

• Step one – Find the slope of a line with points (-4, 3), (-2, 1)

12

2

42

3112

12

mxx

yy

Step Two – Use either point to write the equation in point-slope form. Use (-4, 3)

y – y1 = m(x – x1)

Y – 3 = -1[x – (-4)]Y – 3 = -1(x + 4)

Write in slope-intercept formY – 3 = -1(x + 4)

Y – 3 = -x - 4Y = -x - 1

Writing Equations of Writing Equations of Parallel and Perpendicular Parallel and Perpendicular

LinesLines

Writing Equations of Writing Equations of Parallel and Perpendicular Parallel and Perpendicular

LinesLines

Geometry ConnectionGeometry Connection

In coordinate geometry you studied how to determine if lines where parallel or

perpendicular.

• Nonvertical lines are parallel if they have the same slope and different y-intercept. Any 2 vertical lines are parallel. (y = 3x + 1 and y = 3x -3)

• Two lines are perpendicular if the product of their slopes is -1. A vertical line and a horizontal line are also perpendicular. (y = -¼x 1 and y = 4x + 2)

In the graph on the left, the two lines are parallel. Parallel lines have the same slope and never intersect.

In the graph at the right, the two lines are perpendicular. Perpendicular lines are lines that

intersect to form right angles.

Writing Equations of Parallel Lines

Write the equation for the line that contains (5, 1) and is parallel to y = ¼x – 4.

Identify the slope of the given line. y = ¼x – 4 (slope is ¼)Using point-slope form, write the equation.y – y1 = m(x – x1) point-slope form of an equation

y – 1 = ¼(x – 5) substitute (5, 1) for (x1, y1) and ¼ for m

Y – 1 = ¼x – 5/4 using the Distributive Property, remove the parentheses

Y = ¼x – ¼ simplify and rewrite in slope-intercept

The equation is y = ¼x – 1/4 .

1) Write an equation for the line that contains (2, -6) and is parallel to y = 2x +

9.

2) Write an equation for the line that contains (3, 4) and is parallel to y = ½x –

4.

For perpendicular lines, the product of two numbers is -1, if one number is the

negative reciprocal of the other. Here is how to find the negative reciprocal of a

number.

Start with a fraction:

Find its reciprocal:

Write the negative reciprocal:

Start with an integer:

Find its reciprocal:

Write the negative reciprocal:

5

3

3

5

3

5

4

4

1

4

1

Writing Equations of Perpendicular Lines

Write an equation of the line that contains (0, -2) and is perpendicular to y = 4x + 3.

Identify the slope of the given line. Y = 4x + 3 (slope is 4, the negative reciprocal is –¼).

Using point-slope form, write the equation.y – y1 = m(x – x1) point-slope form if an equation.

y –(-2) = -¼(x – 0) substitute (0, -2) for (x1, y1) and –¼ for m

y + 2 = -¼x – 0 using the Distributive Property, remove the parentheses

Y = -¼x - 2 simplify and rewrite in slope-intercept.

The equation is y = -¼x – 2.

1) Write an equation of the line that contains (1, 8) and is perpendicular to y = ¾x + 1.

2) Write an equation of the line that contains (6, 2) and is perpendicular to y = -2x + 7

Graphing Absolute Graphing Absolute Value EquationsValue Equations

Graphing Absolute Graphing Absolute Value EquationsValue Equations

Distance from zero?Distance from zero?

A V-shaped graph that points upward or downward is the graph of an absolute

value equation

• The Absolute value of a number is its distance from 0 on a number line.

• Make a take of values and graph the equation y = |x| + 1

Below are the graphs of y = |x| + 1 and y = |x| + 2. Describe how the graphs are the

same and how they are different.

The graphs are the same shape. The y-intercept of the first graph is 1. the y-intercept of the second graph is 2

y = |x| + 1 y = |x| +

2

Describe how each graph below is like y = |x| and how it is different.

y = |x| + 3

y = |x| - 3

Graph each equation (function)

1) y = |x| + 2

2) y = |x| – 4

3) y = |x| + 1

4) y = |x| - 5Use x-values of -2, -1, 0, 1, 2

Equation Forms Equation Forms (review)(review)

Equation Forms Equation Forms (review)(review)

When working with straight lines, When working with straight lines, there are often many ways to arrive there are often many ways to arrive

at an equation or a graph.at an equation or a graph.

Slope Intercept Form

If you know the slope and where the line crosses the y-axis, use this form.

y = mx + b

m = slopeb = y-intercept

(where the line crosses the y-axis)

Point Slope Form

If you know a point and the slope, use this form.

y – y1 = m(x – x1)

m = slope

(x1, y1) = a point on the line

Horizontal Lines

y = 3 (or any number)

Lines that are horizontal have a slope of zero. They have “run” but no “rise”. The rise/run

formula for slope always equals zero since rise = o.

y = mx + by = 0x + 3

y = 3This equation also describes what is happening to

the y-coordinates on the line. In this case, they are always 3.

Vertical Lines

x = -2Lines that are vertical have no slope

(it does not exist).They have “rise”, but no “run”. The rise/run

formula for slope always has a zero denominator and is undefined.

These lines are described by what is happening to their x-coordinates. In this example, the x-

coordinates are always equal to -2.

There are several ways to graph a straight line given its equation.

Let’s quickly refresh our memories on equations of straight lines:

Slope-intercept Point-slope Horizontal line Vertical line y = mx + b

When stated in “y=” f orm, it quickly gives

the slope, m, and where the line

crosses the y-axis, b, called the y-intercept.

y - y1 = m(x – x1) when graphing, put this equation into “y=” form to easily

read graphing information.

Y = 3 (or any #) Horizontal lines have

a slope of zero – they have “run”, but no “rise” – all of the

y values are 3.

X = -2 (or any #) Vertical line have no

slope (it does not exist) – they have

“rise”, but no “run” –all of the x values

are -2.

Remember

If a point lies on a line, its coordinates make

the equation true.(2, 1) on the line y = 2x -3 because

1 = 2(2) - 3

Before graphing a line, be sure that your

equation starts with “y =”

To graph 6x + 2y = 8 rewrite the equation:

2y = -6x + 8Y = -3x + 4

Now graph the line using either slope

intercept method or table method.

Practice with Practice with Equations of LinesEquations of Lines

Practice with Practice with Equations of LinesEquations of Lines

Writing and graphing linesWriting and graphing lines

Practice with Equations of LinesAnswer the following questions dealing with

equations and graphs of straight lines.

1) Which of the following equations passes through the points (2, 1) and (5, -2)?

a. y = 3/7x + 5 b. y = -x + 3c. y = -x + 2 d. y = -1/3x + 3

2) Does the graph of the straight line with slope of 2 and y-intercept of

3 pass through the point (5, 13)?

Yes

No

3) The slope of this line is 3/2?

True

False

4) What is the slope of the line3x + 2y = 12?

a) 3b) 3/2c) -3/2d) 2

5) Which is the slope of the line through (-2, 3) and (4, -5)?

a) -4/3b) -3/4c) 4/3d) -1/3

6) What is the slope of the line shown in the chart below?

X 1 3 5 7

Y 2 5 8 11

a) 1b) 3/2c) 3d) 3/5

7) Does the line 2y + x = 7 pass through the point (1, 3)?

True

False

8) Which is the equation of a line whose slope is undefined?

a) x = -5b) y = 7c) x = yd) x + y = 0

9) Which is the equation of a line that passes through (2, 5) and has

slope -3?

a) y = -3x – 3b) y = -3x +

17c) y = -3x +

11d) y = -3x + 5

10) Which of these equations represents a line parallel to the

line 2x + y = 6?

a) Y = 2x + 3b) Y – 2x = 4c) 2x – y = 8d) Y = -2x + 1