In this series, we discuss how to solve systems of equations. This is an important skill for the GRE since many of the math problems will involve solving systems of equations (e.g. problems with inequalities, algebra problems, or word problems). Now, this content builds on some other mathematical ideas, like the idea of a variable and the distributive law. If you want a refresher on those ideas, see this post.

What is a “system of equations”?

A system of equations is a group of one or more equations, like this:

    \[x^2 + 6x + 9 = 0\]

Or this:

    \[\begin{cases} 4x + y =& 8 \\ 2x - y =& 4 \end{cases}\]

And we want to find the possible value(s) of x and any other variables in the problem. So, for example, in the second pair of equations we could have:

    \[x = 2, y = 0\]

And then we would get:

    \[\begin{cases} 4x + y = 4(2) +0 &= 8 \\ 2x - y = 2(2)- 0 &= 4\end{cases}\]

making both equations true. But if we had x = 3, y = 2 then we would get:

    \[\begin{cases} 4x + y = 4(3) +2 &= 8 \\ 2x - y = 2(3)- 2 &= 4\end{cases}\]

which would make the second equation true, but not the first. A solution to a system of equations is an assignment of values to every variable such that every equation in the system is true. By what we have said above, x = 2, y = 0 is a solution to the above pair of equations, whereas x = 3, y = 2 is not.

In this series, we will focus on solving two basic kinds of problems:

  1. Linear Systems of Equations: These only have variables raised to the 1st power. For example:
    • x + y = 21
    • \frac{42}{9} x + 2y + 81 = 13
  2. Quadratic Systems of Equations: These include variables raised to the 2nd power. For example:
    • x^2 = 2
    • 4x^2 + 2x + 6 = 13

What Kinds of Problems Involve This Skill?

This skill can come in handy for many different types of GRE problems, such as:

  • Quantity Comparison Problems
    Suppose x > 100. Which of the following is larger?


    x + 9,900
  • Word Problems
    Tom and Liz are baking cookies. Tom can bake 10 cookies in an hour and Liz can bake 5 cookies in an hour. Working together, how long will it take them to bake 40 cookies?
  • Problems with Infinite Series
    Suppose we have the series: 1, ½, ¼, …
    What is its sum?
  • Problems with Inequalities
    Which values of x, y will satisfy the following system of inequalities?

        \[\begin{cases} x + 3y \ge  & 4 \\ 2x - y < & 1 \end{cases}\]

In our next post, we discuss what a linear equation is, a precursor to actually solving them

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