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# Balancing Scales

What is the weight of the green triangle?

In this quiz, we focus on a technique called **the elimination method.** This important technique helps us to solve both balance puzzles and systems of equations that contain more than one variable.

Given the two balance scales, what is the weight of the green triangle?

If you did not read or understand the solution to the last question, we will explain the solution in more detail here.

While some balance problems can be solved using guess and check, we can also use a more systematic elimination approach.

We are trying to determine the weight of a green triangle.

If we know that a square and a triangle combine to create 3, then we can remove one square, one triangle, and 3 from the second balance.

Now we can see that the blue square equals 2, so the green triangle must equal 1.

Which is greater, the weight of one red circle or the weight of one blue square?

What is the weight of one red circle?

What is the weight of one blue block plus one red circle? (Note that the while objects with positive weights push down on a balance scale, objects with negative weights can be thought of as pulling up on a scale. Therefore, the red circle balloons have negative values.)

NOTE: The red circles have the same magnitude of weight in both scales; when they are balloons the only difference is the value is negative.

What is the weight of one green triangle?

Let's look back on a couple of our balance puzzles and use algebra to solve them.

This process that we have been using to solve a system of equations, or multiple equations with multiple variables, is called **elimination.** Here is the algebraic version of elimination.

If one squares and two circles balance with 9, and three squares and two circles balance with 11, then we know the following:

In this case, we can subtract the first equation from the second one.

Once we know that \(S = 1,\) we can determine that \(C = 4.\)

This process that we have been using to solve a system of equations, or multiple equations with multiple variables, is called elimination. By adding or subtracting two equations, we are able to eliminate a variable from an equation, therefore creating a new equation that is possible to solve.

Which is larger, \(x\) or \(y\,?\)

\[\begin{align} 3x + y &= 9 \\ x - y &= 7 \end{align}\]

In order to solve this system of equations using elimination, which strategy, or strategies, would work for eliminating a variable? \[\begin{align} 3x - 2y &= -5 \\ 2x + 4y &= 18 \end{align}\]

**A.** Multiply the top equation by 2, then add the equations.

**B.** Multiply the top equation by -2, then add the equations.

**C.** Multiply the top equation by 2, then subtract the equations.

**D.** Multiply the top equation by -2, then subtract the equations.

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