Given the equation $2q + 3r = 12$ and $q = 3$, substitute $q = 3$ into the equation:
SEO Optimized Article: Solving the Equation $2q + 3r = 12$ with Given $q = 3$ – A Step-by-Step Guide
When working with linear equations, substitution is one of the most powerful and practical techniques—especially when one variable’s value is known. In this article, we’ll explore solving the equation $2q + 3r = 12$ by substituting $q = 3$, a foundational skill in algebra that applies across mathematics, physics, and engineering.
How to Substitute $q = 3$ into $2q + 3r = 12$
The equation $2q + 3r = 12$ represents a linear relationship between variables $q$ and $r$. If we are given that $q = 3$, the strategy is simple: replace $q$ with $3$ in the equation and solve for $r$.
Step 1: Substitute $q = 3$
Start with the original equation:
$$ 2q + 3r = 12 $$
Replace $q$ with $3$:
$$ 2(3) + 3r = 12 $$
Step 2: Simplify the equation
$$ 6 + 3r = 12 $$
Step 3: Isolate the term with $r$
Subtract $6$ from both sides:
$$ 3r = 12 - 6 $$ $$ 3r = 6 $$
Step 4: Solve for $r$
Divide both sides by $3$:
$$ r = rac{6}{3} = 2 $$
Final Solution
So, when $2q + 3r = 12$ and $q = 3$, we find:
$$ r = 2 $$
This solution gives us the complete ordered pair $(q, r) = (3, 2)$, a clear and verified result.
Why Substitution Matters in Algebra
Substitution is essential not only for solving equations but also for applying systems of equations, modeling real-world problems, and simplifying complex expressions. Mastering this technique strengthens mathematical problem-solving skills used in advanced topics such as calculus, economics, and data science.
Conclusion
Solving linear equations through substitution is efficient and logical. Always verify your work by plugging known values back into the original equation—ensuring accuracy and building confidence. Whether you're studying for an exam or tackling real-world problems, mastering this method is key to success in algebra and beyond.
Key Takeaways:
- Always substitute known values into the equation first.
- Simplify step-by-step to avoid errors.
- Verify solutions by back-substitution.
Start mastering substitution today—your math skills will grow faster than you think!
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