A **system of **linear equations is just a set of two or more linear equations.

In two variables (*x *and *y*), the graph of a system of two equations is a pair of lines in the plane.

There are three possibilities:

- The lines intersect at zero points. (The lines are parallel.)
- The lines intersect at exactly one point. (Most cases.)
- The lines intersect at infinitely many points. (The two equations represent the same line.)

**How to Solve a System Using ****The Substitution Method**

**Step 1:**First, solve one linear equation for*y*in terms of*x*.**Step 2:**Then substitute that expression for*y*in the other linear equation. You'll get an equation in*x*.**Step 3:**Solve this, and you have the*x*-coordinate of the intersection.**Step 4:**Then plug in*x*to either equation to find the corresponding*y*-coordinate.

**Note:** If it's easier, you can start by solving an equation for *x* in terms of *y*, also – same difference!

**Example:**

Solve the system

Solve the second equation for *y*.

* y* = 19 – 7*x*

Substitute 19 – 7*x* for *y* in the first equation and solve for *x*.

3*x* + 2(19 – 7*x*) = 16

3*x* + 38 – 14*x* = 16

–11*x* = –22

* x* = 2

Substitute 2 for *x* in *y* = 19 – 7*x* and solve for *y*.

* y* = 19 – 7(2)

* y* = 5

**Note 2: **If the lines are parallel, your *x*-terms will cancel in step 2, and you will get an impossible equation, something like 0 = 3.

**Note 3: **If the two equations represent the same line, everything will cancel in step 2, and you will get a redundant equation, 0 = 0.