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How many solutions does the equation \(x+y=12\) have?

Which of these graphs is a representation of the equation \(2x+y=5?\)

On the graph below, you can modify the values of the equation. Notice how the graph changes as the equation changes. Every point on the line represents a valid solution to the equation.

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How many solutions satisfy both of the equations below at the same time?

You can use the graphic below to visualize the equations and their relationship to one another. While you won't be able to create this exact problem below, you should be able to gain enough information to answer the question: \[\begin{align} y &= -x + 8\\ y &= x -6. \end{align}\]

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Note: When we are trying to solve multiple equations at the same time, they are called *simultaneous equations* or a *system of equations*.

Under what conditions would two simultaneous linear equations have no solution?

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We saw in the previous problem that equations that make parallel lines have no solution. Which of the following pairs of equations have no solution?

**Note:** You can use the visualization to find which lines are parallel, but there is a faster method that is algebraic.

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How many solutions satisfy the simultaneous equations below? \[\begin{align} x + y &= 3\\ 2x + 2y &= 6 \quad \end{align}\]

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In the previous questions, we've seen that a single equation with two variables does not have a unique solution but instead has many solutions, which we can represent graphically.

In addition, we've seen that when we have two variables and two different equations, this sometime produces a single solution, but sometimes produces no solutions or infinitely many; these possibilities correspond to intersecting lines, parallel lines, and overlapping lines, respectively.

In the following problems, we'll try applying these ideas to some balance problems.

How many values are possible for the weight of the blue square?

How many values are possible for the weight of the blue square?

How many values are possible for the weight of the green triangle?

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