Algebra I

Graphing Equations

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?

Graphing Equations

Which graph shows the solution set for this system of equations? \[ \begin{align} 3x - 4y &= 6 \\ -9x+12y &=-18\end{align}\]

Graphing Equations

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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Graphing Equations

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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Graphing Equations

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.

Graphing Equations

If squares are \(x,\) triangles are \(y,\) and circles are \(z,\) then we get this system of equations: \[\begin{align} x + y &= z \\ 2y &= 2x + z \\3x &= y. \end{align}\] Is it possible to determine the weight of one circle?

Graphing Equations

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.

Graphing Equations

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

Graphing Equations

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

Graphing Equations

What will the graph of the system of equations below look like? \[\begin{align} 3x + 1y + 8z &= 4\\ 9x + 2y + 3z &= 4\\ 1x + 8y - 2z &= 4 \end{align}\]

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