5-3, 5-4, and 5-5 Forms of Linear Equations

Quick Review

The graph of a linear equation is a line. You can write a linear equation in different forms.

The slope-intercept form of a linear equation is y = m x + b , y equals m x plus b comma  where m is the slope and b is the y-intercept.

The point-slope form of a linear equation is y − y 1 = m ( x − x 1 ) , y minus , y sub 1 , equals m open x minus , x sub 1 , close comma  where m is the slope and ( x 1 , y 1 ) open , x sub 1 , comma , y sub 1 , close  is a point on the line.

The standard form of a linear equation is A x + B y = C , eh x plus b y equals c comma  where A, B, and C are real numbers, and A and B are not both zero.

Example

What is an equation of the line that has slope − 4 negative 4  and passes through the point ( − 1 , 7 ) ? open negative 1 comma 7 close question mark

y − y 1 = m ( x − x 1 ) Use point-slope form . y − 7 = − 4 ( x − ( − 1 ) ) Substitute  ( − 1 , 7 )  for  ( x 1 , y 1 )  and − 4  for  m . y − 7 = − 4 ( x + 1 ) Simplify inside grouping symbols . table with 3 rows and 3 columns , row1 column 1 , y minus , y sub 1 , column 2 equals m . open . x minus , x sub 1 . close , column 3 cap usepointminusslopeform . . , row2 column 1 , y minus 7 , column 2 equals negative 4 . open . x minus . open , negative 1 , close . close , column 3 cap substitute . open , negative 1 comma 7 , close . for . open . x sub 1 , comma , y sub 1 . close . and , minus 4 , for , m . , row3 column 1 , y minus 7 , column 2 equals negative 4 . open , x plus 1 , close , column 3 cap simplifyinsidegroupingsymbols . . , end table

An equation of the line is y − 7 = − 4 ( x + 1 ) . y minus 7 equals negative 4 open x plus 1 close .

Exercises

Write an equation in slope-intercept form of the line that passes through the given points.

18. ( − 3 , 4 ) , ( 1 , 4 ) open negative 3 comma 4 close comma open 1 comma 4 close
19. ( 3 , − 2 ) , ( 6 , 1 ) open 3 comma negative 2 close comma open 6 comma 1 close

Write an equation of each line.

Graph each equation.

22. y = 4 x − 3 y equals 4 x minus 3
23. y = 2 y equals 2
24. y + 3 = 2 ( x − 1 ) y plus 3 equals 2 open x minus 1 close
25. x + 4 y = 10 x plus 4 y equals 10

5-6 Parallel and Perpendicular Lines

Quick Review

Parallel lines are lines in the same plane that never intersect. Two lines are perpendicular if they intersect to form right angles.

Example

Are the graphs of y = 4 3 x + 5  and  y = − 3 4 x + 2 bold italic y equals , 4 thirds , bold italic x plus 5 , and , bold italic y equals , negative , 3 fourths , bold italic x plus 2   parallel, perpendicular, or neither? Explain.

The slope of the graph of y = 4 3 x + 5 y equals , 4 thirds , x plus 5  is 4 3 . 4 thirds , .

The slope of the graph of y = − 3 4 x + 2 y equals negative , 3 fourths , x plus 2  is − 3 4 . negative , 3 fourths , .

4 3 ( − 3 4 ) = − 1 4 thirds , open negative , 3 fourths , close equals negative 1

The slopes are opposite reciprocals, so the graphs are perpendicular.

Exercises

Write an equation of the line that passes through the given point and is parallel to the graph of the given equation.

26. ( 2 , − 1 ) ; y = 5 x − 2 open 2 comma negative 1 close semicolon y equals 5 x minus 2
27. ( 0 , − 5 ) ; y = 9 x open 0 comma negative 5 close semicolon y equals 9 x

Determine whether the graphs of the two equations are parallel, perpendicular, or neither. Explain.

28. y = 6 x + 2 y equals 6 x plus 2

    18 x − 3 y = 15 18 x minus 3 y equals 15

29. 2 x − 5 y = 0 2 x minus 5 y equals 0

    y + 3 = 5 2 x y plus 3 equals , 5 halves , x

Write an equation of the line that passes through the given point and is perpendicular to the graph of the given equation.

30. (3, 5); y = − 3 x + 7 y equals negative 3 x plus 7
31. (4, 10); y = 8 x − 1 y equals 8 x minus 1

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