Math Problem Statement
Solve the system of equations: 4x - 7y = -29 and 5x + 6y = 8.
Solution
The problem involves solving the system of equations:
4x - 7y &= -29 \quad \text{(Equation 1)} \\ 5x + 6y &= 8 \quad \text{(Equation 2)} \end{aligned}$$ ### Solution: We will use the substitution or elimination method. Here, I’ll solve using **elimination**: --- #### Step 1: Align and scale the equations for elimination. To eliminate $$y$$, we need to make the coefficients of $$y$$ equal. The least common multiple of 7 and 6 is 42. So: - Multiply Equation 1 by $$6$$: $$6(4x - 7y) = 6(-29) \quad \Rightarrow \quad 24x - 42y = -174$$ - Multiply Equation 2 by $$7$$: $$7(5x + 6y) = 7(8) \quad \Rightarrow \quad 35x + 42y = 56$$ We now have: $$\begin{aligned} 24x - 42y &= -174 \quad \text{(Scaled Equation 1)} \\ 35x + 42y &= 56 \quad \text{(Scaled Equation 2)} \end{aligned}$$ --- #### Step 2: Add the equations to eliminate $$y$$: $$(24x - 42y) + (35x + 42y) = -174 + 56$$ $$59x = -118$$ $$x = -2$$ --- #### Step 3: Substitute $$x = -2$$ into one of the original equations. Using Equation 1 ($$4x - 7y = -29$$): $$4(-2) - 7y = -29$$ $$-8 - 7y = -29$$ $$-7y = -21$$ $$y = 3$$ --- ### Final Answer: $$x = -2, \; y = 3$$ --- Would you like further clarification or additional examples? Here are some related questions: 1. How would you solve this system using substitution instead of elimination? 2. What is the graphical interpretation of this system of equations? 3. How can this system of equations be represented as a matrix? 4. Can you verify the solution by substituting into both equations? 5. What are the advantages of elimination vs substitution for solving systems? **Tip**: Always verify your solution by substituting the values of $$x$$ and $$y$$ back into the original equations!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Equations
Elimination Method
Formulas
4x - 7y = -29
5x + 6y = 8
Theorems
Linear system of equations
Elimination method for solving linear systems
Suitable Grade Level
Grades 8-10
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