11.4+Linear+Equations

Section 11.4 - Linear Equations

 * A linear equation is an equation (equals sign) where the answer, or solution, is actually a line. The equation actually has many possible answers, but they all happen to lie in a straight line when you put them on a graph. **


 * Linear equations have two variables **, which we usually call X and Y. Sometimes the equations don't look like they have both variables, but they really do. For example, if you had 0X, you would have nothing and so you wouldn't even include the X. 3X = 0Y + 4 is the same as 3X = 4. We don't write the 0Y, but we could have added 0Y and not changed the equation.

When you have an equation with a single variable, you would simply solve for that variable. If we have something like 4X + 4 = 16, you would subtract 4 from both sides and divide by the coefficient (4) to find that X = 3.

However, linear equations have two variables and it isn't possible to figure out one variable unless you know the other. The good news is that you can pretty much choose whatever you want for one of the variables and then solve for the other.


 * Paired Data as Solutions**

Consider the equation: 3X = 5Y + 2

Would (4,2) be a solution? Plug the first value in for X and the second for Y and see if the equation would be true? Would (-6,-4) be a valid solution?

Both of these paired data options would be solutions, making the equation true. There are actually an unlimited number of solutions that would make the equation true, and an unlimited number that would not be solutions.

Sometimes you are given one of the values (X or Y) and asked to come up with the value of the other variable that would make the equation true. To do that, simply plug in the value you are given and solve for the other.

What would the value of Y be if X = 7 in the following equation: 4Y = 2X - 2 If you plug in 7 for X in the equation and solve for Y, you find that Y = 3, so (7,3) is the ordered pair that would be one valid solution for this equation.

Solutions do not need to be integers (no fractions or decimals), but it is easier if they are.


 * Table of Values**

Usually in linear equation problems, you are free to choose your own X or Y value and then solve for the other variable. To do this, we use a tool called a table of values.



The table of values helps us keep track of the values for X and Y in an equation, and it allows us to come up with at least three solutions (ordered pairs or paired data) that make the equation true.

The table of values above means "try X = 3 and solve for Y, then try X = 0 and solve for Y, then try Y = 0 and solve for X"

After plugging in the value shown for the specified variable and solving for the other variable, you would add that to the appropriate box in the table of values.



Once you have the values in the table, you would just translate them to ordered pairs or paired data and write them in parentheses, separated by a comma.


 * Graph the Points and Connect the Dots**

Once you have ordered pairs, graph those three points. Then draw a straight line that passes through all three points. If a straight line does not pass through all three points, you did something wrong in figuring out the points. The line does not actually start or end at any point -- it goes on forever in both directions -- so add an arrow to both ends of the line to suggest that.




 * Horizontal and Vertical Lines**

These equations tend to be the tricky ones, because the lack of an X or Y value throws people off. Don't let it. Add 0X or 0Y if you want to make the equation look more normal.

X = - 5

X is always going to equal negative 5. It doesn't matter what Y is. Y could be zero, but it could also be any other number -- X is still negative five, no matter what.



The result is a vertical line.

In the same way, an equation like 4Y = 12 would produce a horizontal line at Y = 3.


 * Chapter Eleven Practice Problems**

**Where to from here?**
11.1 Paired Data 11.2 Like a Grid 11.3 Points 11.4 Linear Equations Chapter Eleven Summary Chapter Eleven Homework