Solving Linear Equations with One Variable

2

To solve a linear equation in one variable:

1. Simplify both sides of the equation.

2. Use the addition and subtraction properties to get all variable terms on the left-hand side and all constant terms on the right-hand side.

3. Simplify both sides of the equation.

4. Divide both sides of the equation by the coefficient of the variable.

Example: Solve x + 1 = 3(x 5).

x + 1 = 3(x 5)

x + 1 = 3x 15

x = 3x 16

2x = 16

x = 8 The solution is 8. Check the solution:

Original equation

Simplify right-hand side.

Subtract 1 from both sides.

Subtract 3x from both sides.

Divide both sides by 2.

True(8) + 1 = 3((8) 5) 9 = 3(3)

3

Equations with fractions can be simplified by multiplying both sides by a common denominator.

The lowest common denominator of all fractions in the equation is 6.

3x + 4 = 2x + 8 3x = 2x + 4

x = 4

Multiply everything on each side by 6.

Simplify.

Subtract 4.

Subtract 2x.

Check.

True

Example: Solve .)4(3

1

3

2

2

1 xx

4)) ((3

1

3

2) (

2

14 4

)8(3

1

3

22

3

8

3

8

)4(

3

1

3

2

2

1xx6 6

4

Alice has a coin purse containing $5.40 in 10 cents coins and 20 cents coins. There are 30 coins all together. How many 10 cents coins are in the coin purse?

Let the number of 10 cents coins in the coin purse = d. Then the number of 20 cents coins = 30 d.

10d + 20(30 d) = 540 Linear equation

10d + 600 20d = 540

10d 20d = 60

10d = 60

d = 6

Simplify left-hand side.

Subtract 480.

Simplify right-hand side.

Divide by 10.

There are 6, 10 cents coins in Alice’s coin purse.

5

The sum of three consecutive integers is 54. What are the three integers?

Three consecutive integers can be represented as n, n + 1, n + 2.

n + (n + 1) + (n + 2) = 54

3n + 3 = 54

3n = 51

n = 17

Simplify left-hand side.

Subtract 3.

Divide by 3.

The three consecutive integers are 17, 18, and 19.

17 + 18 + 19 = 54. Check.

Linear equation

x and y -intercepts

●The x-intercept is the point where a line crosses the x-axis.The general form of the x-intercept is (x, 0).

The y-coordinate will always be zero.

●The y-intercept is the point where a line crosses the y-axis.The general form of the y-intercept is (0, y).

The x-coordinate will always be zero.

To summarize….

●To find the x-intercept, plug in 0 for y.

●To find the y-intercept, plug in 0 for x.

Find the x and y- interceptsof x = 4y – 5

● x-intercept:

● Plug in y = 0

x = 4y - 5

x = 4(0) - 5

x = 0 - 5

x = -5

● (-5, 0) is the

x-intercept

● y-intercept:

● Plug in x = 0

x = 4y - 5

0 = 4y - 5

5 = 4y

= y

● (0, ) is the

y-intercept

5

4

5

4

ExampleQuestion: Draw the graph of 2x + y = 4

Solution

x = 0

2(0) + y = 4

y = 4

1st Co-ordinate = (0,4)

y = 0

2x + 0 = 4

2x = 4

x = 2

2nd Co-ordinate = (2,0)

So the graph will look like this.

y

x 1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

–7 –6 –5 –4 –3 –2 –1 -1-2-3-4-5-6

2x + y = 4

Simultaneous Equations

What are they?• Simply 2 equations

– With 2 unknowns– Usually x and y

• To SOLVE the equations means we find values of x and y that– Satisfy BOTH equations [work in]– At same time [simultaneously]

Elimination Method

2x – y = 1

3x + y = 9

We have the same

number of y’s in each

If we ADD the equations, the y’s disappear+

5x = 10 Divide both sides by 5

x = 2

AB

Substitute x = 2 in equation A2 x 2 – y = 1

4 – y = 1y = 3

Answer

x = 2, y = 3

What if NOT same number of x’s or y’s?

5x + 2y = 173x + y = 10

-x = 3

In B

AB

5 x 3 + 2y = 1715 + 2y = 17

y = 1

Answer

x = 3, y = 1

If we multiply A by 2 we

get 2y in each

5x + 2y = 176x + 2y = 20

B

A

15

Make ‘y’ the subject in 1.

Substitute 1a into equation 2.

5x + 2 = 7

EquationsLinear Simultaneous Equations

Substitution Example

2x - y = -1 1.

x + 2y = 7 2.

y = 2x + 1 1a.

5x = 5 x = 1

x + 2(2x + 1) = 7

Sub x = 1 into 2.

1 + 2y = 7

2y = 6

y = 3

Solution (1, 3)

y

Slope of a Line

• Slope of a line:

rise

run

2 2,x y

1 1,x y

run

rise

xx

yym

12

12

Slope-Intercept Form

The equation

y = mx+b is called the slope-intercept form of an

equation of a line.

The letter m represents the slope and b represents the y-intercept.

Point-Slope Form

1 1( )y y m x x

2 1

2 1

y ym

x x

Cross-multiply and

substitute the more

general x for x2

where m is the slope and (x1, y1) is a given point.

It is derived from the definition of the slope of a line:

The point-slope form of the equation of a line is

ExampleFind the equation of the line through the points (–5, 7) and (4, 16).

Example

19

9

)5(4

716

m

Now use the point-slope form with m = 1 and (x1, x2) = (4, 16).

(We could just as well have used (–5, 7)).

Find the equation of the line through the points (–5, 7) and (4, 16).

Solution:

12164

)4(116

xxy

xy

Distance between two points

The formula

Why?

• The formula, derived from the Pythagoras theorem states that square a and square b, add them together, and square root them, you get the length of c, hypotenuse.

ExampleGiven the coordinates of two points we can use the formula

to directly find the distance between them. For example:

What is the distance between the points A(5, –1) and B(–4, 5)?

A(5, –1) B(–4, 5)x1 x2y1 y2

( ) ( )x x y y 2 22 1 2 1+

2 2 2 2( 4 5) +(5 1) = ( 9) + 6

= 81+36= 117 = 3 13

The Mid-Point of a LineIn general, the coordinates of the mid-point of the line segment

joining (x1, y1) and (x2, y2) are given by:

,1 2 1 2+ +

2 2

x x y y

is the mean of the x-

coordinates.

x x1 2+

2

is the mean of the y-

coordinates.

y y1 2+

2

(x2, y2)

(x1, y1)

x

y

0

,1 2 1 2+ +

2 2

x x y y

Example:

END

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