If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Algebra 1

### Course: Algebra 1>Unit 4

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 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?
• The way my teacher writes functions is O(I)=R
O = the output usually y
I = The input variable usually x
R = The functions rule to compute a situation