Linear Equations

Linear Equations Linear Equations

Identify linear equations and functions.

Write linear equations in standard form and graph them.

1) linear equation2) linear function3) standard form4) y-intercept5) x-intercept

Sara has 4 hours after dinner to study and dohomework. She has brought home algebra 2and chemistry.


y

x

6

0

4-2-2

2

6

-2 6

0

-2

6

2

4

4 yx


If she spends x hours on algebra and y hours on chemistry, a portion of the graphof the equation x + y = 4 can be used to relate how much time she spends on each.


y

x

6

0

4-2-2

2

6

-2 6

0

-2

6

2

4

4 yx


If she spends x hours on algebra and y hours on chemistry, a portion of the graphof the equation x + y = 4 can be used to relate how much time she spends on each.

An equation such as x + y = 4 is called a linear equation.


A linear equation has no operations other than:

________, ___________, and ____________________________________.



________, ___________, and ____________________________________.addition



________, ___________, and ____________________________________.addition subtraction



________, ___________, and ____________________________________.addition subtraction multiplication of a variable by a constant




The variables may not be multiplied together or appear in a ____________.




The variables may not be multiplied together or appear in a ____________.denominator





A linear equation does not contain variables with exponents other than __.

denominator





A linear equation does not contain variables with exponents other than __.1

denominator






denominator

The graph of a linear equation is always a ____.






denominator

The graph of a linear equation is always a ____.line






denominator


Linear Equations

735 yx

9x

1536 ts

xy21






denominator


Linear Equations Not Linear Equations

735 yx

9x

1536 ts

xy21

847 2 ba

5 xy

1 xyx

xy

1


A linear function is a function whose ordered pairs satisfy a linear equation.



Any linear function can be written in the form f(x) = mx + b, where m and bare real numbers.




State whether each function is a linear function. Explain.

a) f(x) = 10 – 5x





a) f(x) = 10 – 5x Yes!





a) f(x) = 10 – 5x Yes!

It can be written as f(x) = – 5x + 10m = – 5, b = 10

b) g(x) = x4 – 5





a) f(x) = 10 – 5x Yes!


b) g(x) = x4 – 5 No!





a) f(x) = 10 – 5x Yes!


b) g(x) = x4 – 5 No!

x has an exponent other than 1.





a) f(x) = 10 – 5x Yes!


b) g(x) = x4 – 5 No!


c) h(x, y) = 2xy





a) f(x) = 10 – 5x Yes!


b) g(x) = x4 – 5 No!


c) h(x, y) = 2xy No!





a) f(x) = 10 – 5x Yes!


b) g(x) = x4 – 5 No!


c) h(x, y) = 2xy No!

Two variables are multiplied together.


To avoid decompressionsickness, it is recommendedthat divers ascend no fasterthan 30 feet per minute.





The linear function P(d) = 62.5d + 217can be used to find the pressure (lb/ft2)at a depth of d ft. below the surface ofthe water.





Find the pressure at a depth of 350 feet.






P(d) = 62.5d + 217






P(d) = 62.5d + 217

P(350) = 62.5(350) + 217






P(d) = 62.5d + 217

P(350) = 62.5(350) + 217

P(350) = 22,092






P(d) = 62.5d + 217

P(350) = 62.5(350) + 217

P(350) = 22,092

The pressure at a depth of 350 feet is about 2 000,22ftlb


Any linear function can be written in standard form.



Standard Form of a Linear Function




The standard form of a linear function is Ax + By = C, where A > 0,





A and B are not both zero.






Also, A, B, and C are integers whose greatest common factor is 1







Write the equation in Standard Form:

43 xy







43 xy

43 yx







62

1

2

1 yx







62

1

2

1 yx

12 yx

In the previous lesson, you graphed functions by using a table of values.



Since two points determine a line, there are quicker ways to graph linear functions.




One way is to find the points at which the graph intersects each axis and then connectthem with a line.

y

x





y

x

The y-coordinate of the point at which the graphcrosses the y-axis is called the ____________.


(0, 3)




y

x

The y-coordinate of the point at which the graphcrosses the y-axis is called the ____________.y - intercept

(0, 3)





y

x


The x-coordinate of the point at which the graphcrosses the x-axis is called the ____________.

(0, 3)

(-4, 0)





y

x


The x-coordinate of the point at which the graphcrosses the x-axis is called the ____________.x - intercept

(0, 3)

(-4, 0)


Find the x-intercept and the y-intercept of the graph of the equation.Then graph the equation.

y

x


1535 yx


y

x

The x-intercept is the value of x when y = 0.


1535 yx

15)0(35 x


y

x



1535 yx

15)0(35 x

155 x


y

x


(3, 0)


1535 yx

15)0(35 x

155 x

3x


y

x


(3, 0)


1535 yx

15)0(35 x

155 x

3x

The y-intercept is the value of y when x = 0.

153)0(5 y


y

x

The x-intercept is the value of x when y = 0.(0, 5)

(3, 0)


1535 yx

15)0(35 x

155 x

3x


153)0(5 y

153 y

5y


y

x


(3, 0)


1535 yx

15)0(35 x

155 x

3x


153)0(5 y

153 y


y

x


(3, 0)


1535 yx

15)0(35 x

155 x

3x


153)0(5 y

153 y

5y

(0, 5)


y

x

The x-intercept is the value of x when y = 0.(0, 5)

(3, 0)


1535 yx

15)0(35 x

155 x

3x


153)0(5 y

153 y

5y

Linear Equations

Technology

Transcript of Linear Equations