including olympiad champions, researchers, and professionals.

including olympiad champions, researchers, and professionals.

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A group of equations that we want to solve at the same time is called a *system of equations*.

Which of the following could represent a system of equations with no solution?

How many solutions does the following system of equations have? \[ \begin{align} y - x &= 2 \\ y + x &= 6 \\ 2y - 2x &= 4 \end{align}\] (The first and second equations have already been graphed for you in red and green, respectively.)

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What values of \(a\) and \(b\) will produce a single solution to the system of equations below? \[ \begin{align} x - 2y &= 3 \\ x + 3y &= 8 \\ x + ay &= b \end{align}\]

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In a previous quiz, we saw that for linear equations with two variables the solution set can be represented as a straight line:

If we have two or more linear equations, the system can be represented by multiple lines. The solution set of the system of equations is the common intersection of these lines. If there is no common intersection, the system of equations has no solution.

The same process can be applied to equations with more variables.

What is the weight of a circle?

One way to think about equations with three variables (like those in the last two problems) is that a linear equation in three variables describes a plane instead of a line.

When we reason about a system of linear equations in three variables, instead of looking for intersections of *lines*, we look for intersections of *planes* to find the solution set.

Which of the following could represent a system of equations with no solution?

Which of the following represents a system of equations with infinite solutions?

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