*A graph can be used to show the solution for a system of two linear equations.However, accurately determining the solution from a graph is not always easy or accurate.*

*A graph can be used to show the solution for a system of two linear equations.However, accurately determining the solution from a graph is not always easy or accurate.*

In cases like this, you can use algebraic methods to find exact answers.

One method to look at is called the substitution method.

Take note of how we have an equation with variables on both sides. Pay close attention to the very last step of the solution.

Let's take a look at another example where you will find that you cannot solve the system.

We knew, from the previous lesson, that this system represents two parallel lines.

But I tried, by substitution, to find the intersection point anyway. Since there wasn't any intersection point, my attempt led to utter nonsense."), just as two identical lines are quite different from two parallel lines. A useless result means a dependent system which has a solution (the whole line); a nonsense result means an inconsistent system which has no solution of any kind.In the examples above, one of the equations was already given to us in terms of the variable x or y.This allowed us to quickly substitute that value into the other equation and solve for one of the unknowns.In this case, there are an infinite number of solutions. Parallel lines have the same slope, but she also has to check whether they have different y-intercepts because the lines could be collinear (remember that 2 collinear lines are the same line).If Aubrey finds that the slopes of the lines are the same and the y-intercepts are different, then she can be confident that her answer is correct. The origin has no bearing on whether two lines are parallel.But in a dependent system, the "second" equation is really just another copy of the first equation, and all the points on the one line will work in the other line." — something that's true, but unhelpful (I mean, duh! Neither of these equations is particularly easier than the other for solving.I'll get fractions, no matter which equation and which variable I choose. I guess I'll take the first equation, and I'll solve it for, um, Keep in mind that, when solving, you're trying to find where the lines intersect. Then you're going to get some kind of wrong answer when you assume that there is a solution (as I did when I tried to find that solution).The method of solving "by substitution" works by solving one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging this back into the other equation, "substituting" for the chosen variable and solving for the other. The idea here is to solve one of the equations for one of the variables, and plug this into the other equation.It does not matter which equation or which variable you pick.For example, where do you think the two lines shown below intersect?It looks like they might intersect at (1.8, –0.7)—though this is only an estimate.

## Comments How To Solve A Substitution Problem

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