Big Ideas Math Algebra 2 A Bridge to Success
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Big Ideas Math Algebra 2 A Bridge to Success View details
Cumulative Assessment

Exercise 1 Page 644

a Our friend claims that any system formed by three of the following equations has exactly one solution.
Equation

We want to find a system of three equations that supports his claim. For example, we can pick the following three equations.

Equation
To check whether a system of the three highlighted equations has only one solution, we need to solve it.
To do this we will use the Elimination Method. Notice that the coefficients of the variables and in Equation (I) are the same as in Equation (III).
Let's subtract Equation (III) from Equation (I) to eliminate variables and
Simplify
We found that Let's substitute that for in Equations (II) and (III).
Equations (II) and (III) form a system of two equations with two variables.
At this point we can solve it as we would with any other systems of two equations. First, let's simplify them.
Simplify
We found that To find the value of we can substitute into the second equation.
We solved the system of equations and found only one solution: and As we were solving the system we did not stumble upon any identities like or contradictions such as Therefore, this system of equations has only one solution and it supports our friends claim.
b This time, we want to show that our friend's claim is incorrect. To do that, we want to find a system that does not have only one solution.
Equation

For example, we can pick the following three equations.

Equation
To check whether a system of the three highlighted equations has only one solution, we need to solve it.
To do this we will use the Elimination Method. Notice that the coefficients of the variables and in Equation (II) are the same as in Equation (III).
Let's subtract Equation (III) from Equation (II) to eliminate variables and
Simplify
We found that Let's substitute for in Equations (I) and (III).
Equations (I) and (III) form a system of two equations with two variables.
At this point we can solve it as we would do with any other systems of two equations. First, let's simplify them.
Simplify

& Add and subtract terms

We obtained an identity, This means that this system has infinitely many solutions. Let's try to describe them. We know that and that
Therefore, any ordered triple of the form is a solution to the system to the three selected equations. This means that our friend's claim is incorrect.