linear equations
- Thread starter gullpacha
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lev888
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Which points in the coordinate plane make the equation true? Any points with y=3. Such points form a line. It would've been a point if the x coordinate was constrained to one value. It is not.
(0,3) is a point just like (1,3) is a point. 0 is a real number just as valid as another number. it's not a placeholder as in ( • , 3 )
But you are right there is some confusion because it depends on the context (ambient space). x=1 is a point on the line (x-axis), but if we are in 2-dimensions then x=1 specifies the line (1,y). A point in the plane would have both x and y specified ex. x=1, y=0.
But you are right there is some confusion because it depends on the context (ambient space). x=1 is a point on the line (x-axis), but if we are in 2-dimensions then x=1 specifies the line (1,y). A point in the plane would have both x and y specified ex. x=1, y=0.
Cubist
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Sorry that was a bad pun! I was making the joke that the words "plane" and "plain" sound similar.
A linear equation in 2D space (like y=3, or y=2x+1) will reduce the available space for x and y from the whole XY plane onto a line. Adding another linear equation will usually further reduce the available space from the line down to a singe point (unless the second equation specifies the same line that the first equation did). The remaining point will be at the intersection of the two lines.
When you go further in math, you'll find out that a linear equation will also reduce a 3D volume to a 2D plane. That was the basis of my silly joke!
Steven G
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Lets not confuse the beginning algebra student.
In x+3 = 4 we have x=1. It is not a line, it is not the point (1,0). It is the solution to an equation.
The line x=3 is a line, it contains many points. In fact it contains each and every point on the x-y plane (just think graph paper) where the x value is 1. Like (1, 3/4), (1, 7), (1, -11), (1, 3sqrt(5)), ...
x=1 never represents the point (1,0). After all, why not (1,3)?
You need to understand where x=1 is coming from. If I ask you if x=1 is a solution to ...., then x is only one. It is not a point it is just a possible solution to an equation.
Same goes for y= some constant.
In x+3 = 4 we have x=1. It is not a line, it is not the point (1,0). It is the solution to an equation.
The line x=3 is a line, it contains many points. In fact it contains each and every point on the x-y plane (just think graph paper) where the x value is 1. Like (1, 3/4), (1, 7), (1, -11), (1, 3sqrt(5)), ...
x=1 never represents the point (1,0). After all, why not (1,3)?
You need to understand where x=1 is coming from. If I ask you if x=1 is a solution to ...., then x is only one. It is not a point it is just a possible solution to an equation.
Same goes for y= some constant.