General MathematicsADE 101

Unit 2LECTURE No. 14

TYPES OF LINEAR EQUATIONS

Today’s Objectives•Understand linear equations and its types.

•Form the linear equations involving slopes of different situations.

Students and

Teachers will be able to

Knowledge Test

Point – Slope FormTo write an equation of a line in point – slope form, all you need is …

… Any Point On The Line …

… The Slope …

(x1, y1)

m

Once you have these two things, you can write the equation as

y – y1 = m (x – x1)

That’s “y minus the y-value of the point equals the slope times the quantity of x minus the x-value of the point”.

ExampleWrite the equation of the line that goes through the point (2, –3) and has a slope of 4.

Point = (2, –3)

Slope = 4

y – y1 = m (x – x1)

y + 3 = 4 (x – 2)

Starting with the point – slope form

Plug in the y-value, the slope, and the x-value to get

Notice, that when you subtracted the “–3” it became “+3”.

y – y1 = m (x – x1)

Starting with the point – slope form

Plug in the y-value, the slope, and the x-value to get

Notice, that when you subtracted the “–4” it became “+4”.

Write the equation of the line that goes through the point (–4, 6) and has a slope of . 3

2Point = (–4, 6)

Slope =

32

y – 6 = (x + 4)32

Example

Write the equation of the line that goes through the points (6, –4) and (2, 8) .

Point = (6, –4)

Slope = –3y + 4 = –3 (x – 6)

We have two points, but we’re missing the slope. Using the formula for slope, we can find the slope to be

( )8 4 12 32 6 4

m- -

= = =-- -

Point = (2, 8)

Slope = –3

To use point – slope form, we need a point and a slope. Since we have two points, just pick one … IT DOESN’T MATTER … BOTH answers are acceptable… more on why later.

y – 8 = –3 (x – 2)

Using the first point, we have, Using the second point, we have,

y2 – y1

x2 – x1

Example

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)

Writing Equations in Slope – Intercept Form

y + 4 = –3 (x – 6) y – 8 = –3 (x – 2)

Earlier we wrote an equation of the line that went through the points (6, –4) and (2, 8) . Sometimes, we want the line written in a different form.To change a point-slope equation in slope-intercept form, solve for y and simplify the right side of the equation.

- Solve for y: Add or subtract the y-value of the point to both sides

- Simplify: Distribute the slope and then combine like terms.Here are the two answers we had from the earlier example.

Subtract 4 from both sides Add 8 to both sides

y = –3 (x – 6) – 4 y = –3 (x – 2) + 8

SIMPLIFYDistribute –3 and combine like terms

Distribute –3 and combine like terms

y = –3x + 18 – 4

y = –3x + 14

y = –3x + 6 + 8

y = –3x + 14Notice … They’re the same!

Write the equation of the line in slope-intercept form that goes through the point (6, 2)

and has slope .

23

Begin in point-slope form: y – 2 = (x – 6)23

y = (x – 6) + 223

23

Distribute: y = x – 4 + 2

Combine Like Terms: y = x – 223

Add 2 to both sides

Solve for y:

Example

y as a function of xFor an equation to be written as a function, you must solve for y.

Solving for y means that “y is written as a function of x ”.

When your equation is in point – slope form simply add or subtract the y-value of the point to the other side.

y + 3 = 4 (x – 2)

From our first example we had

In order to write y as a function of x we subtract 3 from both sides of the equation.

y = 4 (x – 2) – 3

When you write y as a function of x, you have put your equation in function form.

You may replace the y with the notation f (x) … read “f of x ” or “function of x ”. f (x) = 4 (x – 2) – 3

Of the three types of linear equations discussed in this presentation, only slope-intercept form is written as a function.

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

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 FormWriting an equation when you know a point

(2, 5) and the slope m = 2

Other Forms of Linear EquationsSo far, we have discussed only point-slope form. There are other forms of equations that you should be able to identify as a line and graph if necessary. Horizontal Line: y = c , where c is a constant.Vertical Line: x = c , where c is a constant.

Slope – Intercept Form: y = mx + b

Standard Form: Ax + By = C

To write equations in the last two forms, start in point – slope form and rearrange the variables to match the correct format.

The next few slides will cover how to do this.

m = the slope of the line … b = the y-intercept

Example: y = 3

Example: x = –6

Example: y = 3x – 6

A, B, and C are integers.

Example: 3x + 4y = –36

Writing Equations In Standard FormThe last form of a linear equation we are going to cover is called Standard Form.

Ax + By = C , where A, B, and C are integers.

If you needed to write an equation of a line in standard form, you would start in point-slope form or slope-intercept form, depending on what information you are given.

In both cases, you must put all variables on the left side and all constant values on the right side.

If any of the coefficients (A, B, or C) are NOT integers, then you must eliminate any fractions or decimals by multiplying every term in the equation by the appropriate factor.

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 + b

Y = 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

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.

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 non-vertical 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 exampleUsing 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