1.6 Systems of two linear inequalities 

SAT Math has at least one question (but good chance more than that) on setting up and solving a system of linear inequalities.  This article walks you through solving a system of linear inequalities.

A system of two or more linear inequalities is made up of two linear inequalities that cross somewhere on the X-Y plane and the solution to these equations is the point where the two lines cross.

Two lines cross at the point (2,1) according to the graph.

Let’s find this solution algebraically, using the equations that are provided for each line.

So we have two equations:

x-y=1 and

x+y=3

We have two equations and two unknowns.  Since the number of equations is the same as the number of unknowns, we can solve this system.

First, let’s express one variable in terms of another variable.

We’ll take the first equation, x-y=1 and express x in terms of y.

We’ll add y to both sides in order to isolate the x on the left side.

x – y + y = 1 + y

-y and +y cancel each other out on the left side, leaving us with

x = 1 + y

Secondly, we’ll plug the first equation that we just rearranged into the second equation x+y=3.

We will substitute x in x+y=3 with 1+y, because that’s what x is equal to in the first equation.

After substituting, we get

1+y+y=3

Adding two y’s together and subtracting 1 from both sides, we get

2y=2

Dividing both sides by 2 gives us

y=1

Thirdly, we go ahead and plug 1 for y into the first equation [x=1+y] in order to find x

x=1+1

x=2

So we found solution for the cross point algebraically, it is (2,1).

Does it correspond to the graphical solution?  Let’s look up at the graph.

The graph shows that the lines cross at the point (2,1), so both solutions correspond to one another.

Practice Problem Video

x + 3y = 3

x – y = 2

Try doing it by yourself first and watch the video to check your answers or to walk you through the solution. (Coming Soon)

 

More Practice Problems

Here are some more problems for you to try.  Solve the following systems of equations.

In case you are running into any kind of difficulties with this topic or any other SAT Math topic, feel free to sign up for a one-on-one session with me.  I’m available to tutor SAT Math online or in person for Tampa, Brandon, Riverview and other Tampa Bay area students.  Happy studying!