To determine the best solution to the system using the substitution method rather than graphing, let's analyze each of the provided solutions in detail. The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation.
Given solutions:
a. [tex]\((2, -5)\)[/tex]
b. [tex]\((0, 0)\)[/tex]
c. [tex]\(\left(\frac{6}{11}, -\frac{9}{11}\right)\)[/tex]
d. [tex]\(\left(-10, -\frac{1}{2}\right)\)[/tex]
Evaluating the Solutions:
1. Solution (2, -5):
- This solution involves positive and negative integers, which could indicate that it may not be the simplest solution when substituting, as large numbers and different signs can complicate the solving process.
2. Solution [tex]\(\left(\frac{6}{11}, -\frac{9}{11}\right)\)[/tex]:
- This solution involves fractions, making the substitution method more complex with the need to manage fractional arithmetic.
3. Solution (0, 0):
- This solution involves zero values for both variables. Substituting zero is straightforward and often results in simpler algebraic expressions. Since zero nullifies terms involving multiplication, the equations become much simpler.
4. Solution [tex]\(\left(-10, -\frac{1}{2}\right)\)[/tex]:
- This solution involves a mix of a large negative integer and a negative fraction, which introduces complexity in substitutions due to dealing with signs and mixed types of numbers (integers and fractions).
Upon analyzing all the solutions, we can see that [tex]\((0, 0)\)[/tex] is the simplest and most straightforward when substituted back into the system of equations. Substituting zero for any variable generally simplifies the equations considerably, making this solution ideal for solving the system using the substitution method.
Therefore, the best solution found by the substitution method over graphing is:
[tex]\[
(b) \, (0, 0)
\][/tex]
