Math Problem Statement
Solution
Let's solve the given system of equations using the substitution method. The equations are:
Step 1: Substitute into the second equation
From the first equation, is expressed in terms of :
Substitute this into the second equation, :
Step 2: Simplify the equation
Expand :
Combine like terms:
Convert to a fraction with a denominator of 8:
So the equation becomes:
Step 3: Solve for
Subtract 5 from both sides:
Multiply through by 8:
Step 4: Solve for
Substitute into the first equation, :
Simplify:
Final Answer:
The solution is:
Would you like further details on any step? Here are related questions for practice:
- How would you solve this system of equations using the elimination method?
- What happens if one equation cannot be expressed explicitly for substitution?
- Can systems of equations always have a single solution? Why or why not?
- What does it mean if the equations are dependent or inconsistent?
- How can you graphically interpret the solution to this system?
Tip: Always simplify fractions and verify by substituting the solution back into the original equations.
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Linear Equations
Substitution Method
Formulas
y = mx + b for substitution
Simplification of linear equations
Theorems
Properties of Equality
Linear Equation Principles
Suitable Grade Level
Grades 8-10
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