february 17, 2015
TRANSCRIPT
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Parallel &Perpendicular Lines
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Parallel Lines Have the Same Slope
• The lines never touch. Therefore:• There is no solution to a system of equations• If you know the slope of one line, you know the
slope of a line parallel to it.• If you know the slope and a point on the line,
you can use the point-slope formula to find the equation of, and graph the line.
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The question says: The equation of a line parallel to the one shown could be;
It doesn’t matter what points the line goes through, as long as the slope is the same
Find the slope..
A) y = 3x – 7 B) y < 3x – 7 C) 4x +2y = -11 D) 3x –y = 4
Which is the equation of a line parallel to all those shown on the graph?
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Write the equation and graph the line parallel to the one shown and passing through the point (3,-4).
Whole different story, this one. The line must pass thru
Again, find the slope of the current line, then use the point-slope formula to find the new equation.
Point-Slope:Y + 4 = -3(x – 3)
Equation:Y = -3x + 5
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A. y = -4/3x + 3 B. y = 4/3x -2 C. y = -3/4x + 2 D. y = -4x -3
E. None
An equation of a line parallel to the graph could be:
Parallel Lines
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What is the equation of the line parallel to the line
2y –x = 1, and passing thru the point (-4,5)
A. y = -x + 5/2 B. y = 2x – 5/2 C. y = -1/2x + 5
D. y = 1/2x – 5 E. None
Parallel Lines
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Perpendicular Lines Have Slopes that are the opposite inverse of each other
• The lines cross at a 90 degree angle
• There is always one solution
• If you know the slope of one line, change the sign and use the reciprocal
• If you know the slope and a point on the line, you can use the point-slope formula to find the equation of, and graph the line.
Write the equation ofthe line perpendicularto the one given
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Write the equation of the line perpendicular to the one shown passing through the point (3,-4).
Again, find the slope of the current line, then use the point-slope formula to find the new equation.
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Graph the inequality: y < 5x + 1
Graphing Systems of Inequalities(3)
Let’s start by graphing an inequality:
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Class Notes: Systems of Inequalities(3)
1. Write the equation in slope-intercept form.
2. Graph the y-intercept and slope.
3. Draw the line (solid or dashed).
, Dashed line
, Solid line
• Steps to Graphing Linear System Inequalities
, Above y-intercept
, Below y-intercept
4. Lightly shade above or below the y-intercept.
5. Graph the other equation. See #’s 3 and 4
6. Darkly shade overlap.
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21
3
45
3
y x
y x
Ex.
Graph the system of linear inequalities.
2) Graph.
Find m and b.
3) Solid or
dashed?
4) Lightly
shade above
or below the
y-intercept?
1) Put in
slope-intercept
form.
5) Do the same
for the other
equation.
6) Darkly
shade overlap.
2
3m 1b
4
3m 5b
Solid Below
Dashed
Above
Class Notes: Systems of Inequalities
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15
2
3 2
y x
y x
Graph the system of linear inequalities.
2) Find m and
b, then graph
3) Solid or
dashed?
4) Lightly
shade above
or below the
y-intercept?
1) Put in slope-
intercept form.
5) Do the same
for the other
equation.
6) Darkly
shade overlap.
1
2m 5b
3
1m 2b
Dashed Above
Dashed
Above
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Solving 3x3 Systems of Equations
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Solving 3x3 Systems
A 3x3 system of equations has 3 unknown variables, and therefore must have 3 equations.
We will look at two methods of solving 3x3 systems. The method used depends entirely on the number of unknowns in each equation.
A) Only 1 of the equations has all three variables in the equation. This is the easier of the two. Let’s look:
Solve the System: 4x + 2y - z = -5 3y + z = -1
2z = 10
Begin at the bottom and work your way up.
Plug z into 2nd equation and solve for y.
Plug y and z into 1st
equation and solve for x.Plug all three in together and check your solutions.
5. The solution set is (1, -2, 5)
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Solving 3x3 Systems
B) All 3 of the equations contain all three variables in the equation. Follow these steps to solve:
Steps for Solving in 3 Variables1. Take the 1st 2 equations, cancel one of the variables.
2. Take the last 2 equations, cancel the same variable from step 1.
3. Take the results from steps 1 & 2 and use elimination solve for both variables.
4. Plug the results from step 3 into one of the original 3 equations and solve for the 3rd remaining variable.
5. Write the solution as an ordered triple (x,y,z).
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1. Solve the system.
(2, -4, 1)
3 11
2 1
5 2 3 21
x y z
x y z
x y z
Solving 3x3 Systems
x + 3y – z = -11+ 2x + y + z = 1
2x + y + z = 1+ 5x – 2y + 3z = 21
Must eliminate the z here also.
3x + 4y = -10 - x - 5y = 18
+ -x - 5y = 183( )
+ -3x -15y = 54
- 11y = 44
-3( )
y = - 4
3x + 4(-4) = -10
3x = 6x = 2
2 + 3(- 4) – z = -11
– z = -1z = 1
Plug all three into one of original equations to check.
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Class Work 3.4:
Show all work on separate sheet of paper.
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