DECKS 1&2 COMBINED Flashcards
What is a solution to an equation?
A solution is the set of values that makes an equation TRUE.
How can you tell if the point (5, -3) is a solution to a linear equation?
If it “works.” Sub 5 in for the X and -3 in for the Y. Simplify. If the statement is true, then it is a solution. If it is a solution, then it is on the line.
What are the slopes of the diagonals of a square?
+1 going up L to R, and -1 going down from L to ROW
What is the equation of the y axis?
x=0
Describe what the line y=x looks line
positive diagonal line from bottom left to upper right, slope of 1, through the origin
How can you think of the points on the line y=-x to help you graph it?
the x values are opposite the y values… so (2, -2) and (-2, 2) and (-107, 107).
What is the slope formula you use when trying to find the slope if you are given two points?
m = ( y - y ) / ( x - x )
What is the slope of a vertical line?
undefined
when you see “ - y” or “ - x” what can you think of?
opposite of y and opposite of x
Parallel lines have ______ slopes, or slopes
the same
y = 1/2 x - 8 what is the slope?
1/2
If a line as a slope of 2/3, then any line parallel to it will have a slope of ___ and any line perpendicular to it will have a slope of ___
parallel 2/3 and perp -3/2
y=mx+b what form?
slope intercept
what does a slope of zero look like?
horizontal, flat from left to right. The x axis has a slope of zero.
If you are given two points and need to find the equation of a line, how do you do it?
First, fine the slope using m=(y-y)/(x-x), then use that slope and EITHER OF THE POINTS. Put it in point slope form.
what does a slope that is undefined look like?
vertical, straight up and down. The y axis has a slope of zero.
What is the equation of the x axis?
y=0
What is the x intercept of a line? (what are the coordinates?)
Where the line crosses the horizontal axis, the x axis. The coordinates are ( x, 0 )
How do a lot of people think about the slope (the slope)?
rise/run… or change in y / change in x
The three forms of equations of a line we use are :
point-slope, slope-intercept, standard
what is the origin?
the point (0,0)
perpendicular lines have _____
opposite AND reciprocal slopes
(switch sign and flip the fraction)
What do you know about all Y intercepts?
(0,y) they have an x coordinate of ZERO
Why is it called “slope-intercept” form?
There is a slope and an intercept explicitly shown in the formula
if a slope of zero is 0 degrees, what is a slope of -1?
-45 degrees downhill (traveling L to R)
How can you find the slope if it is in standard form?
Solve for y and change it into slope intercept form.
parallel lines have ______
the same slope
How can you find the x intercepts from an equation of a line?
Plug zero in for Y, and solve for X. The coordinates will be ( X, 0)
What is the “point slope” form?
y-y = m (x-x) (has a point and a slope in it)
How can you tell if the point (5, -3) is on a line if you are given the equation?
If it “works.” Sub 5 in for the X and -3 in for the Y. Simplify. If the statement is true, then it is a solution. If it is a solution, then it is on the line.
how can you think of the points on the line y=x to help you graph it?
all of the x values are equal to all of the y values.. so (2,2), (3,3), (5,5), (-100,-100) are all in the solution set and on the line
What is the graph of a line showing us?
It is showing us all of the points, (x,y), that make the equation true. Every dot on the graphed line will make the equation true, they all work.
what is the “most popular” form?
slope intercept y=mx+b
When you graph a line, what is it?
It is a bunch of points!!! all have coordinates (x,y) it is the solution set, all of the coordinates that WORK in the equation.
4x - 7y = 28 what is x intercept?
plug 0 in for y, 28/4=8, so (8,0)
What do you know about all X-intercepts?
they have a y coordinate of ZERO (x,0)
If we are given a Y value, like y=8, and need to find the x that goes with it, how do you do it?
substitute the 8 in for the y in the equation and solve for x. That x goes with the 8. the point (x, 8) will be on the line.
what are the coordinates of a point?
(x,y), the x and y value.
what is point-slope form a rearrangement of?
A rearrangement of the slope formula: y-y =m (x-x)
describe what the line x=-8 looks like
vertical, straight up and down, through the point (-8,0) on the x axis. undefined slope
If a slope of zero is 0 degrees, what is a slope of 1?
45 degrees uphill (traveling L to R)
What does “vertical” mean?
up and down
What is an interesting fact about slopes between -1 and +1?
HALF OF ALL POSSIBLE SLOPES ARE BETWEEN -1 and +1 (and they are perpendicular to eachother)
What is the slope of a line?
it is the steepness
How can you think of a slope of 1 or -1?
If you draw a square, think of the diagonals.. 45 degrees!
If I take a point (x,y) that is NOT on the line and plug it into the equation what will happen?
It will make a false statement (the sides won’t be equal). It will look like: 5=2 or 123 = -8
the number 0/5 is equal to :
zero. 0
the number 5/0 is equal to :
undefined
If a line as a slope of -5, then any line parallel to it will have a slope of ___ and any line perpendicular to it will have a slope of ___
-5 and perp 1/5
When you graph a line, what are you actually graphing?
THE SOLUTION! Those are all of the points that make the equation true. All of the points that “work”. If you put any other pionts, (x,y) into the equation, they wont work, You will get a false statement.
If an x intercept is 455, what point is on the line?
(455, 0)
If you draw a square with the diagonals as an X, what is the slope of the diagonal that goes from top left to the bottom right corner?
-1
y - 7 = 3 ( x + 9) what form?
point slope [there is a point (-9,5) and a slope of 3]
What is your favorite class?
ALGEBRA 2
y = 1/2 x - 8 what point is given?
(0, -8) , the y intercept
If we are given an X, like x=5, and need to fine a Y that goes with it, how do we find it?
substitute the 5 in for the x in the eqution and solve for y, That y value goes with the 5. The point (5, y) will be on the line.
Why is it called “point-slope” form?
There is a point and a slope explicitly shown in the form
What is the y intercept of a line? (what are the coordinates?)
Where the line crosses the vertical axis, the y axis. The coordinates are ( 0, Y )
How can you think of points on the line y=5 to help you graph it?
all of the points on the cartesian plane with a y value of 5… (2,5), (9,5), (-200,5)
what does a zero slope look like?
left to right (horiz)
If I take a point (x,y) that is on the line and plug it into the equation, what will happen?
It will make a true mathematical statement (the left and the right side will be equal). It will look like 6=6 or 0=0 or -1.8 = -1.8
y-y= m (x-x) what form?
point slope
What is the eqution of a horizontal line that goes througyh the y axis at 8? How can you think about it to help you?
y = 8. Since y is the height, you can think of it as all of the points that have a height of 8. Also, you can think of it as all of the points that have a y value of 8, so you are a solution if your y value is 8.
what is an undefined slope look like?
up and down (vert)
How can you think of the line x=6 to help you graph it?
all of the points on the cartesian plane with an x value of 6: (6, 0), (6, -3), (6, 24.3), (6, -2000)
4x - 7y = 28 what is x intercept?
plug 0 in for x, 28/-7=-4, so (0,-4)
y - 7 = 3 ( x + 9) what point is given?
(-9, 7) it is in point slope form
describe what the line y=12 looks like
horizontal, from left to right, through (0,12)
what is the cartesian plane?
it is the thing we plot points on with the x y axis.. named after RENE DESCARTES, french philosopher and mathematician.
What is the slope-intercept form?
y = mx + b
What does it mean to “satisfy” an equation?
Make the equation true.
If a y intercept is 990, what point is on the line?
(0, 990)
Perpendicular lines have _____ slopes, or slopes
OPPOSITE SIGN AND RECIPROCAL
What is the “standard” form?
ax + by = c no fractions or decimals and “a” is positive
what is the XY axis graph thing we plot lines and points on called?
the cartesian plane
How can you always get rid of fractions in any equation?
Multiply every term by the LCD (plop the LCD in the numerators, then simplify and all of the fractions VANISH!!!!)
Why can’t you divide by zero? (why can zero be in the top and not the bottom?)
10/2 asks “how many times do you add 2 to itself to get to 10?” the answer is 5.10/0 asks “how many times do you add 0 to istelf to get to 10?” the answer is there is no answer. undefined
How can you find the y intercepts from an equation of a line?
Plug zero in for X and solve for Y. The coordinates wll be (0, Y)
What is the slope of a horizontal line?
ZERO
y - 7 = 3 ( x + 9) what is the slope?
3
How do you show that a point lies on a line?
Show that, when you plug the point in, you get a true mathematical statement.
If you draw a square with the diagonals as an X, what is the slope of the diagonal that goes from bottom left to upper right corner?
1
y = 1/2 x - 8 what form?
slope intercept
4x - 7y = 28 what form?
standard
What does “horizontal” mean?
left to right.. like a floor or table top
What is the x coordinate for all y-intercepts?
zero. all y intercepts are in the middle, not left or right, because they are on the y axis.
what is the point (0,0) called?
the origin
Describe what the line y=-x looks like
negative diagonal line from top left to bottom right, slope of -1, through the origin
What is the y coordinate for all x-intercepts?
zero. all x intercepts have a height of zero because they are on the x axis.
what is the slope of a line that goes straight up and down?
undefined
ax + by = c what form?
standard
What is the slope of a flat horizontal surface?
ZERO
What is the equation of a vertical line that goes through the x axis at 7? How can you think of that line to help make sense of it?
x = 7. You can think “all of the points on this line have an x value of 7, so you are a solution as long as your x=7.
If you are asked to find the equation passing through a given point with a given slope, what is the best form to use?
If you are given a point and a slope, use POINT SLOPE!! y-y=m(x-x) OR y [opp y] = slope (x [opp x])
If you take a point, (x,y), that is NOT on the line and plug it into the equation what will happen?
You get a FALSE statement.
