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### Course: Algebra (all content)>Unit 3

Lesson 1: Two-variable linear equations intro

# Two-variable linear equations intro

Learn about a class of equations in two variables that's called "linear equations." They are called that way because their graph is a line. These are the most basic and probably most useful equations you will ever know!

## Want to join the conversation?

• If x is an irregular number such as pi, can the equation still be linear? Thanks!!
• Yes, it's just harder to graph, as the graph uses increments of whole numbers for the most part, but just imagine a point that's about 3.14 or some other point on the graph that's very close to the result you are looking for.
• I have come from bangladesh to u.s.a and now iam will go to high school 10 grade,, can u tell me what type of math i have to peactice
• So if I just plug a number in the x value, it will lead me to the y value? And then I can just graph it and it will be in line with the other points?
• Find the mid point of line joining the point (22,20) and (0,16)
• what is simutaneos equation
• A Simultaneous Equation is a finite set of equations which has common solutions. It is same as Systems of linear equations.
• Is a linear equation the same thing as a proportional relationship? Don't proportional relationships have to pass through the origin though?
• Yes, a proportional relationship is a linear relationship that goes through the origin, so not all linear relationships are proportional.
• Why does the solution of a linear equation always come out to be a straight line on the graph?
• Good question!

In x and/or y, any linear equation is equivalent to one of two forms: x=a or y=mx+b where a, m, and b are constants. (Yes, this already includes the form where y is a constant, because this would be the result of taking m to be 0 in the equation y=mx+b).

For the form x=a, the graph is the set of all points with constant x-coordinate a. This is clearly a vertical line through (a, 0).

For the form y=mx+b, if we can show that the slope is constant, then the graph must be a straight line. For any two distinct points on the graph, the x-coordinates, say x_1 and x_2, are also distinct (since the same value of x would give the same value of y because y is given explicitly in terms of x). The two points are (x_1, mx_1+b) and (x_2, mx_2+b). The distinctness of x_1 and x_2 avoids division by 0 when we find the slope.

For these two points, the slope is

[(mx_2+b)-(mx_1+b)]/(x_2-x_1)
= (mx_2-mx_1)/(x_2-x_1)
= m(x_2-x_1)/(x_2-x_1)
= m.

The slope is the constant m, so the graph of y=mx+b is a straight line.

We conclude that the graph of any linear equation in x and/or y is always a straight line.
• what is the point in this its not like we are going to need to know this stuff to get into the army?!
• Well, not everybody is in the army and that isn't the only thing you can do in life. You're going to need to know this stuff in order to help advance the world's technology.
• What would a linear equation look like in the First, Third, or even Fourth dimensions?
• A line will still be a line in whatever dimensions you are talking about.
• Is there a function like this:
x=yz
or xa=yz?
And what's the difference between linear equations and functions?