Math Problem Statement
Solve each system of equations
Solve each system of equations
Solution
The system of equations shown in the image is:
1. \quad 2p - q + 4r &= 11 \\ 2. \quad p + 2q - 6r &= -11 \\ 3. \quad 3p - 2q - 10r &= 11 \end{aligned}$$ ### Step 1: Solve one equation for a variable Let's solve the second equation for $$p$$: $$p = -2q + 6r - 11$$ ### Step 2: Substitute $$p$$ in the other equations Now, substitute this expression for $$p$$ into the first and third equations: #### Substituting into the first equation: $$2(-2q + 6r - 11) - q + 4r = 11$$ Simplify: $$-4q + 12r - 22 - q + 4r = 11$$ $$-5q + 16r = 33 \quad \text{(Equation 4)}$$ #### Substituting into the third equation: $$3(-2q + 6r - 11) - 2q - 10r = 11$$ Simplify: $$-6q + 18r - 33 - 2q - 10r = 11$$ $$-8q + 8r = 44 \quad \text{(Equation 5)}$$ ### Step 3: Solve the system of two equations We now have the system: $$\begin{aligned} -5q + 16r &= 33 \\ -8q + 8r &= 44 \end{aligned}$$ Multiply the second equation by 2 to eliminate $$r$$: $$-16q + 16r = 88$$ Subtract the first equation from this: $$(-16q + 16r) - (-5q + 16r) = 88 - 33$$ Simplify: $$-11q = 55$$ $$q = -5$$ ### Step 4: Solve for $$r$$ Substitute $$q = -5$$ into the first reduced equation: $$-5(-5) + 16r = 33$$ $$25 + 16r = 33$$ $$16r = 8 \quad \Rightarrow \quad r = \frac{1}{2}$$ ### Step 5: Solve for $$p$$ Substitute $$q = -5$$ and $$r = \frac{1}{2}$$ into the equation for $$p$$: $$p = -2(-5) + 6\left(\frac{1}{2}\right) - 11$$ $$p = 10 + 3 - 11 = 2$$ ### Final Solution: The solution to the system is: $$p = 2, \quad q = -5, \quad r = \frac{1}{2}$$ Would you like more details or have any questions about this process? Here are some related questions to explore: 1. How do you solve systems of equations using substitution? 2. What other methods can be used to solve a system of equations (e.g., elimination)? 3. How does solving a system with three variables differ from two-variable systems? 4. Why is it important to check solutions in all original equations? 5. Can this system be solved using matrices or determinants? **Tip**: Always substitute your solution back into the original system of equations to verify that it satisfies all the equations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Systems of Equations
Substitution Method
Algebraic Manipulation
Formulas
Substitution for variables
Solving linear equations step-by-step
Theorems
Substitution Method Theorem
Elimination Method
Suitable Grade Level
Grades 9-12
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