3a - 5 = 8 - 2a - jntua results
Solving 3a – 5 = 8 – 2a: A Step-by-Step Guide
Solving 3a – 5 = 8 – 2a: A Step-by-Step Guide
Solving linear equations can seem daunting at first, but with a structured approach, it becomes straightforward. One common problem students encounter is solving equations like 3a – 5 = 8 – 2a. Whether you're a high school student, a homeschooler, or just learning algebra, mastering this type of equation is essential. In this article, we’ll walk through how to solve 3a – 5 = 8 – 2a step by step, explain key algebraic concepts, and provide tips to build confidence in solving similar problems.
Understanding the Context
Understanding the Equation: What Does 3a – 5 = 8 – 2a Mean?
The equation 3a – 5 = 8 – 2a contains an unknown variable a on both sides—and different coefficients (3 and –2) and constants (5 and 8). This type of equation is called a linear equation in one variable, where the variable appears only once to the first power.
Our goal is to isolate a to determine its value. This requires applying algebraic operations to both sides while maintaining equality.
Key Insights
Step-by-Step Solution to 3a – 5 = 8 – 2a
Step 1: Eliminate parentheses and combine like terms
The equation has no parentheses, but we can isolate a by gathering all terms containing the variable on one side and constants on the other.
Start with:
3a – 5 = 8 – 2a
Add 2a to both sides to bring all a terms together:
3a + 2a – 5 = 8
5a – 5 = 8
Now the variable a is on the left, and constants/day coefficients are on the right.
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📰 Lösung: Sei \( d = \gcd(a,b) \). Dann gilt \( a = d \cdot m \) und \( b = d \cdot n \), wobei \( m \) und \( n \) teilerfremde ganze Zahlen sind. Dann gilt \( a + b = d(m+n) = 100 \). Also muss \( d \) ein Teiler von 100 sein. Um \( d \) zu maximieren, minimieren wir \( m+n \), wobei \( m \) und \( n \) teilerfremd sind. Der kleinste mögliche Wert von \( m+n \) mit \( m,n \ge 1 \) und \( \gcd(m,n)=1 \) ist 2 (z. B. \( m=1, n=1 \)). Dann ist \( d = \frac{100}{2} = 50 \). Prüfen: \( a = 50, b = 50 \), \( \gcd(50,50) = 50 \), und \( a+b=100 \). Somit ist 50 erreichbar. Ist ein größerer Wert möglich? Wenn \( d > 50 \), dann \( d \ge 51 \), also \( m+n = \frac{100}{d} \le \frac{100}{51} < 2 \), also \( m+n < 2 \), was unmöglich ist, da \( m,n \ge 1 \). Daher ist der größtmögliche Wert \( \boxed{50} \). 📰 Frage: Wie viele der 150 kleinsten positiven ganzen Zahlen sind kongruent zu 3 (mod 7)? 📰 Lösung: Wir suchen die Anzahl der positiven ganzen Zahlen \( n \le 150 \), sodass \( n \equiv 3 \pmod{7} \). Solche Zahlen haben die Form \( n = 7k + 3 \). Wir benötigen \( 7k + 3 \le 150 \), also \( 7k \le 147 \) → \( k \le 21 \). Da \( k \ge 0 \), reichen \( k = 0, 1, 2, \dots, 21 \), also insgesamt 22 Werte. Somit gibt es \( \boxed{22} \) solche Zahlen. 📰 You Wont Believe What Hidden Gems Pismo Beach Camping Has To Offer This Summer 📰 You Wont Believe What Hidden Mold Lurks In Your Showerpink Compensation 📰 You Wont Believe What Hidden Pokemon Come From This Christmas Advent Calendar 📰 You Wont Believe What Hidden Secrets Are Hidden In Pokmon Black 2 📰 You Wont Believe What Hidden Secrets Are Hidden In Pokmon Black And White 2 📰 You Wont Believe What Hidden Secrets Lurk In Pokmon Black White 2 📰 You Wont Believe What Hidden Secrets Phasmophobia Ps5 Unlockedwatch Now 📰 You Wont Believe What Hides In A Pepper Potts Secret Recipe 📰 You Wont Believe What Led To The Epic Pirates Of Caribbean Film Plot 📰 You Wont Believe What Legends Were Playing On This Playstation 3 📰 You Wont Believe What Lies Inside Pawniardyoull Want To Click Now 📰 You Wont Believe What Lies Inside Pigeon Point Lighthouseshocking History Awaits 📰 You Wont Believe What Lies On The Dead Mans Chestdeadly Treasure Uncovered 📰 You Wont Believe What Makes A Bookcompletethese 7 Parts Will Shock You 📰 You Wont Believe What Makes Poison Weakness Perfectly ObservableFinal Thoughts
Step 2: Isolate the constant terms
Subtract 5 from both sides to eliminate the constant on the left:
5a – 5 – 5 = 8 – 5
5a = 3
Step 3: Solve for a
Now divide both sides by 5 to isolate a:
a = 3 ÷ 5
a = 0.6
Final Answer
✅ The solution to the equation 3a – 5 = 8 – 2a is a = 0.6 (or a = 3/5 in fraction form).
Key Takeaways from Solving 3a – 5 = 8 – 2a
1. Balance is Key
Every operation you perform—adding, subtracting, multiplying—must be applied to both sides. This preserves the equation’s equality.
