Interpreting Systems of Linear Equations Graphically

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  • 00:00 A Linear System of Equations
  • 1:12 Graphing the System
  • 2:02 Finding the Solution
  • 2:57 Example
  • 4:03 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Graphing a system of linear equations is one way to find the solution of the system. Graphing the system out also lets you see if there is no solution at all.

A Linear System of Equations

In this lesson you will see how you can interpret or solve a system of linear equations by graphing it. Knowing how to use this graphing method can help you visually solve such a system without having to use algebra.

Remember, a linear system of equations is a set of linear equations. If you have one variable, then you have one equation. If you have two variables, which is the usual case for problems you will encounter, then you will have two equations. If you have three variables, then you will have three equations, and so on. But, since they are all linear equations, none of the variables will have exponents.

Here's an example of a linear system in two variables. This is the one we will be solving, too: y = 2x - 3, y = -3x + 2. See how each equation has the same two variables, x and y, and because we have these two variables we have two equations, and because this is a linear system none of our variables have exponents?

Right now all our equations are written in slope-intercept form, which is y = mx + b, making it easy for us to graph. If our equations were written in standard form, which is Ax + By = C, then we can rearrange the equations into slope-intercept form to make it easy for us to graph.

Graphing the System

So, on to graphing our two equations. This is what we get. The red line is for the equation y = 2x - 3, and the blue line is for the equation y = -3x + 2. To graph these two lines, we begin with our y intercept, the last number in our equation. For y = 2x - 3 it is -3. For y = -3x + 2 it is 2. The y intercept is where these lines cross the y axis.

After we have this point, then we find our next point by looking at the slope of the equation. For y = 2x - 3 the slope is 2 which means our next point is 2 up and one to the right of our y intercept. We follow the same procedure for the other equation.

Finding the Solution

To find the solution to this system of linear equations we look to see if the lines intersect. If they do, then the point of intersection is the solution. If the lines don't intersect, then there is no solution. But, if you only see one line, then you have an infinite number of solutions.

Looking at our graph, we see that our two lines do cross, we have a solution. What is it?

Because, we have graphed our equations we can easily find the answer by looking at the graph. Where do our two lines meet on the graph?

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