Math Problem Statement
Can you solve all of these linear equations step by step
Solution
Sure, let's solve these linear equations step by step.
Step 1: Solve for , , and
We have the following system of equations for the column player (economy):
1. & \quad 3q_1 - 2q_2 = v \\ 2. & \quad 2q_1 - q_2 = v \\ 3. & \quad q_1 + 5q_2 = v \\ 4. & \quad q_1 + q_2 = 1 \end{align*}$$ First, we solve for $$q_1$$ and $$q_2$$ using the equation $$q_1 + q_2 = 1$$. $$q_2 = 1 - q_1$$ Substitute $$q_2 = 1 - q_1$$ into the first three equations: 1. $$3q_1 - 2(1 - q_1) = v$$ 2. $$2q_1 - (1 - q_1) = v$$ 3. $$q_1 + 5(1 - q_1) = v$$ Simplify these equations: 1. $$3q_1 - 2 + 2q_1 = v \implies 5q_1 - 2 = v$$ 2. $$2q_1 - 1 + q_1 = v \implies 3q_1 - 1 = v$$ 3. $$q_1 + 5 - 5q_1 = v \implies -4q_1 + 5 = v$$ Now, we have: $$\begin{align*} 5q_1 - 2 &= v \quad \text{(Equation 1)} \\ 3q_1 - 1 &= v \quad \text{(Equation 2)} \\ -4q_1 + 5 &= v \quad \text{(Equation 3)} \end{align*}$$ ### Equating Equations 1 and 2: $$5q_1 - 2 = 3q_1 - 1 \implies 2q_1 = 1 \implies q_1 = \frac{1}{2}$$ ### Equating Equations 2 and 3: $$3q_1 - 1 = -4q_1 + 5 \implies 7q_1 = 6 \implies q_1 = \frac{6}{7}$$ Since $$q_1$$ has different values in these cases, the system seems inconsistent. Thus, we have to check the accuracy or look for any possible mistakes in the original equations. Instead, let's use a different method of solving it. ### Step 2: Solve for $$p_1$$, $$p_2$$, $$p_3$$ and $$v$$ Let's use the value $$q_1 = \frac{1}{7}$$ (common fraction) to solve for $$v$$ and other strategies. The simplified equation should be reconsidered to avoid contradictions. Would you like to proceed with this attempt or adjust the equations?Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Systems of Equations
Formulas
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Theorems
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Suitable Grade Level
High School