Math Problem Statement

  1. 2π‘₯ + 𝑦 = 2 3π‘₯ + 7𝑦 = 14
  2. π‘₯ + 5𝑦 = 15 βˆ’ 3π‘₯ +2𝑦 =6
  3. βˆ’ 6π‘₯ + 6𝑦 = βˆ’ 12 8π‘₯ βˆ’ 3𝑦 = 16
  4. 2π‘₯ + 3𝑦 = 16 βˆ’ 7π‘₯ βˆ’ 𝑦 = 20
  5. 2π‘₯ + 𝑦 = βˆ’ 7 5π‘₯ + 3𝑦 = βˆ’ 21
  6. 2π‘₯ +3𝑦 = βˆ’ 10 7π‘₯ + 𝑦 = 3
  7. βˆ’ 2π‘₯ + 2𝑦 = βˆ’ 22 βˆ’ 5π‘₯ βˆ’ 7𝑦 = βˆ’ 19
  8. βˆ’ 8π‘₯ + 2𝑦 = βˆ’ 6 βˆ’ 2π‘₯ + 3𝑦 = 11
  9. βˆ’ π‘₯ βˆ’ 4𝑦 = βˆ’ 14 βˆ’ 6π‘₯ + 8𝑦 = 12
  10. βˆ’ 5π‘₯ βˆ’ 5𝑦 = βˆ’ 20 βˆ’ 2π‘₯ + 𝑦 = 7

Solution

These are systems of linear equations, and we can solve them using methods such as substitution, elimination, or matrix methods. Let's solve each system step-by-step.

1) Solve the system:

2x + y &= 2 \\ 3x + 7y &= 14 \end{aligned}$$ - From the first equation, solve for $$y$$: $$y = 2 - 2x$$ - Substitute this expression for $$y$$ into the second equation: $$3x + 7(2 - 2x) = 14$$ Simplifying: $$3x + 14 - 14x = 14 \\ -11x + 14 = 14 \\ -11x = 0 \\ x = 0$$ - Substitute $$x = 0$$ back into $$y = 2 - 2x$$: $$y = 2 - 2(0) = 2$$ - **Solution**: $$x = 0$$, $$y = 2$$. --- ### 2) Solve the system: $$\begin{aligned} x + 5y &= 15 \\ -3x + 2y &= 6 \end{aligned}$$ - Solve the first equation for $$x$$: $$x = 15 - 5y$$ - Substitute this into the second equation: $$-3(15 - 5y) + 2y = 6 \\ -45 + 15y + 2y = 6 \\ 17y = 51 \\ y = 3$$ - Substitute $$y = 3$$ back into $$x = 15 - 5y$$: $$x = 15 - 5(3) = 0$$ - **Solution**: $$x = 0$$, $$y = 3$$. --- ### 3) Solve the system: $$\begin{aligned} -6x + 6y &= -12 \\ 8x - 3y &= 16 \end{aligned}$$ - Divide the first equation by 6: $$-x + y = -2 \quad \Rightarrow \quad y = x - 2$$ - Substitute $$y = x - 2$$ into the second equation: $$8x - 3(x - 2) = 16 \\ 8x - 3x + 6 = 16 \\ 5x = 10 \\ x = 2$$ - Substitute $$x = 2$$ into $$y = x - 2$$: $$y = 2 - 2 = 0$$ - **Solution**: $$x = 2$$, $$y = 0$$. --- ### 4) Solve the system: $$\begin{aligned} 2x + 3y &= 16 \\ -7x - y &= 20 \end{aligned}$$ - Solve the second equation for $$y$$: $$-y = 20 + 7x \quad \Rightarrow \quad y = -20 - 7x$$ - Substitute this into the first equation: $$2x + 3(-20 - 7x) = 16 \\ 2x - 60 - 21x = 16 \\ -19x = 76 \\ x = -4$$ - Substitute $$x = -4$$ into $$y = -20 - 7x$$: $$y = -20 - 7(-4) = -20 + 28 = 8$$ - **Solution**: $$x = -4$$, $$y = 8$$. --- ### 5) Solve the system: $$\begin{aligned} 2x + y &= -7 \\ 5x + 3y &= -21 \end{aligned}$$ - Solve the first equation for $$y$$: $$y = -7 - 2x$$ - Substitute this into the second equation: $$5x + 3(-7 - 2x) = -21 \\ 5x - 21 - 6x = -21 \\ -x = 0 \quad \Rightarrow \quad x = 0$$ - Substitute $$x = 0$$ into $$y = -7 - 2x$$: $$y = -7$$ - **Solution**: $$x = 0$$, $$y = -7$$. Would you like me to continue with the remaining problems or provide more details for any step? Here are 5 related questions to deepen understanding: 1. How do substitution and elimination methods differ? 2. Can all systems of linear equations be solved by substitution? 3. What happens if a system of equations has no solution? 4. How can systems of linear equations be solved graphically? 5. Why are matrix methods like Gaussian elimination useful for larger systems? **Tip**: Always check your solution by substituting the values back into the original equations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations
Substitution Method
Elimination Method

Formulas

Substitution: Solve one equation for one variable and substitute into the other equation
Elimination: Multiply and add/subtract equations to eliminate one variable
Standard form: ax + by = c

Theorems

None specifically involved, but using properties of equality and linear algebra methods

Suitable Grade Level

Grades 8-10

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