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Level 254

Linear Equations in Two Variables

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These two numbers make up an ordered pair; we write an ordered pair using (parentheses).
We can represent a specific location, or point, in a rectangular coordinate system by using two numbers; an x-value and a y-value.
Quadrants are labeled as follows:
I = Right Upper Quadrant
Points that lie on an axis are not considered to be in any quadrant.
3(3) + 2(5) = 19
Is (3, 5) a solution to 3x + 2y = 19?
3(6) = 2(-4) + 10
Is (6, -4) a solution to 3x = 2y + 10?
1.) Substitute a value for one of the variables.
To find a solution to a linear equation in two variables:
x = 1
Find three solutions to 2x + y = 13
You simply choose it ;-)
Where do you find the value for x?
When solving linear equations in two variables that contains a fraction,
choose multiples of the denominator to end up working with integers.
x + y
Making a table of values can help keep the ordered pairs organized.
To graph a linear equation in two variables:
1.) Find three ordered pairs that are solutions to the linear equation.
1.) Let y = 0
Finding the x-intercept of a linear equation:
1.) Let x = 0
Finding the y-intercept of a linear equation:
3x - 6(0) = 12
x = (4, 0) and y = (0, -2)
To graph an equation using the intercepts:
1.) Find the x-intercept by letting y = 0 and solving for x.
Some linear equations only have one intercept.
Most linear equations have one x-intercept and one y-intercept.
Graphing linear equations through the origin
An equation in the form Ax + By = 0, where A and B are not both zero, will pass through the origin and the only intercept will be (0, 0)
m = rise / run
One way that we can describe the slope of a line on a graph is
(x1, y1) (x2, y2)
The rise is found by subtracting y-values
On a graph, the slope measures the steepness of a line
- A line with a positive slope goes up from left to right.
The slope of a horizontal line is 0.
The slope of a vertical line is undefined.
Remember the formula to find slope is
m = y2 - y1 / x2 - x1
y = -6x
1.) Find the y-intercept.
1.) Find the slope and the y-intercept.
Graphinf a linear equation with slope and y-intercept
What if the point given is not the y-intercept?
We can use another form of a linear equation called point-slope form.
1.) Write the point-slope equation y - y1 = m (x - x1)
The point-slope form of an equation of a line whose slope is m and that passes through the point (x1, y1) is given by y - y1 = m (x - x1)
Finding the slope of a line that passes through (x1, y1) and (x2, y2)
1.) Put it into the formula m = y2 - y1 / x2 - x1