# BM1.8 – Graphing Linear Systems of Equations

Chapter 1, Lesson 8

In this lesson students will:

• calculate the $x$ and $y$ intercepts of a linear equation
• use the intercepts of a linear equation to sketch the linear equation
• interpret the solution of a system of linear equations as the point of intersection of two lines

## Systems of Linear Equations

#### Intercepts of a system of Linear Equations

In this lesson the main goal will be to sketch a system of linear equations in the Cartesian plane ($xy$-plane). There are several methods to do this, but perhaps the simplest one is to find the $x$ and $y$ intercepts of each linear equation. An intercept is where our line will cross the $x$-axis or $y$-axis. If our line crosses the $x$-axis, we call this point on the line the $x$ intercept. If our line crosses the $y$-axis, we call this point on the line the $y$ intercept.

To find and sketch an $x$-intercept:

1. Substitute $y = 0$ into the linear equation.
2. Solve for $x$.
3. For the value that you found in part 2, count that many spaces along the $x$-axis and draw a point.

To find and sketch a $y$-intercept:

1. Substitute $x = 0$ into the linear equation.
2. Solve for $y$.
3. For the value that you obtained in part 2, count that many steps along the $y$-axis and draw a point.

If you connect the $x$ and $y$ intercepts of a given linear equation with a ruler this will give you a rough sketch of the line it represents in the Cartesian plane. The following video will go through an example of how this is done.

#### Interpreting the Point of Intersection

Once you have drawn a rough sketch of your linear system, you might notice that the two lines intersect at a specific point. It turns out that the solution to the linear system that we found algebraically in chapter BM1.7 has a geometric representation as this intersection point. This is a nice way to double-check that the solution we obtained by solving algebraically seems reasonable given our drawing of the two lines. The short video below will connect these two ideas together for you.