What is a solution to an equation?
A solution is the set of values that makes an equation TRUE.
What does it mean to “satisfy” an equation?
It means you made the equation true. Solutions satisfy equations.
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.
what is the XY axis graph thing we plot lines and points on called?
the cartesian plane
What is the cartesian plane?
That x y axis thing we draw and graph points on
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)
What is the slope formula? (the slope formula?)
m = ( y - y ) / ( x - x )
The three forms of equations of a line we use are :
point-slope form, slope-intercept form, general form
Why is it called “point-slope” form?
There is a point and a slope explicitly shown in the form
where does the point-slope form come from?
A rearrangement of the slope formula: rearrange m=(y-y)/(x-x) into y-y =m (x-x)
Why is it called “slope-intercept” form?
There is a slope and a y intercept explicitly shown in the formula
What is the slope-intercept form?
y = mx + b
What is the “standard” form of a linear equation?
ax + by = c no fractions, a must be positive
what is the “most popular” form?
slope intercept
y = mx + b what form is this?
slope intercept
y-y=m(x-x) what form is this?
point slope form
Ax+By=C what form is this?
standard linear form
If you graph a line, then any points, (x,y) on the line will make the equation ___ if you plug in the x and y
TRUE
If you graph a line, then any points, (x,y) NOT on the line will make the equation ___ if you plug in the x and y
FALSE
Solutions to equations are the values that make the 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.
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.
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.
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.
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 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 )
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 )
what are the coordinates of a point?
(x,y), the x and y value.
What is the y coordinate for all x-intercepts?
zero. (x, 0), all x intercepts have a height of zero because they are on the x axis.
What is the x coordinate for all y-intercepts?
zero.(0, y) all y intercepts are in the middle, not left or right, because they are on the y axis.
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)
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 equation of the x axis?
y=0
What is the equation of the y axis?
x=0
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.
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.
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.
Describe what the line y=x looks line
positive diagonal line from bottom left to upper right, slope of 1, through 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
describe what the line y=12 looks like
horizontal, from left to right, through (0,12)
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 an x intercept is 455, what point is on the line?
(455, 0)
If a y intercept is 990, what point is on the line?
(0, 990)
