Question: Solve for $ z $ in the equation $ 3(z - 4) + 7 = 2(4z + 1) $. - Decision Point
Solving for $ z $: A Step-by-Step Guide to Solve $ 3(z - 4) + 7 = 2(4z + 1) $
Solving for $ z $: A Step-by-Step Guide to Solve $ 3(z - 4) + 7 = 2(4z + 1) $
Equations are fundamental in algebra, and solving for a variable like $ z $ often appears in school curricula and real-world problem-solving. Understanding how to isolate $ z $ step-by-step not only builds mathematical proficiency but also builds confidence in tackling more complex equations. This article walks you through solving the equation:
$$
3(z - 4) + 7 = 2(4z + 1)
$$
Understanding the Context
Step 1: Expand Both Sides
Start by expanding the expressions on both sides of the equation:
$$
3(z - 4) + 7 = 2(4z + 1)
$$
Distribute the $ 3 $ and the $ 2 $:
$$
3z - 12 + 7 = 8z + 2
$$
Now combine like terms on the left-hand side:
$$
3z - 5 = 8z + 2
$$
Step 2: Move All Terms Containing $ z $ to One Side
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Key Insights
Subtract $ 3z $ from both sides to get all $ z $ terms on the right:
$$
-5 = 5z + 2
$$
Step 3: Eliminate the Constant from the Right Side
Subtract $ 2 $ from both sides:
$$
-7 = 5z
$$
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Step 4: Solve for $ z $
Divide both sides by $ 5 $:
$$
z = rac{-7}{5}
$$
Final Answer
$$
oxed{z = - rac{7}{5}}
$$
Why This Equation Matters
This seemingly simple equation teaches core algebraic skills: distributing parentheses, combining like terms, isolating variables, and using inverse operations. Mastering this equation supports learning more advanced topics like systems of equations, linear inequalities, and even modeling real-life situations.
