# Equations and Variables

How many solutions does the equation $$x+y=12$$ have?

## Equations and Variables

### Equations and Unknowns

# Equations and Variables

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

## Equations and Variables

### Equations and Unknowns

# Equations and Variables

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.

## Equations and Variables

### Equations and Unknowns

# Equations and Variables

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}

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

## Equations and Variables

### Equations and Unknowns

# Equations and Variables

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

## Equations and Variables

### Equations and Unknowns

Select one or more.

# Equations and Variables

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.

## Equations and Variables

### Equations and Unknowns

Select one or more.

# Equations and Variables

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

## Equations and Variables

### Equations and Unknowns

# Equations and Variables

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.

## Equations and Variables

### Equations and Unknowns

# Equations and Variables

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

## Equations and Variables

### Equations and Unknowns

# Equations and Variables

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

## Equations and Variables

### Equations and Unknowns

# Equations and Variables

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

## Equations and Variables

### Equations and Unknowns

×