Title: Linear Equations in One Unknown – A Complete Guide
Meta Description: Unlock the secrets of linear equations in one unknown. Learn its importance, solve problems, and explore its concepts with examples and solutions. Perfect for students and math lovers.
—
What is a Linear Equation in One Unknown?
A linear equation in one unknown is an algebraic equation that involves only one variable raised to the power of one. It has the general form:
$$ ax + b = 0 $$
Where:
– \( x \) is the unknown variable
– \( a \) and \( b \) are constants and \( a \neq 0 \)
The solution to the equation is the value of \( x \) that satisfies the equation. These equations form the basis of algebra and are used in solving problems across Mathematics and real-life situations.
Studying this topic is crucial for developing logical thinking and is foundational for advanced math concepts. Top mathematicians like Terence Tao and Maryna Viazovska have recently studied and contributed to linear algebra applications, further showing the relevance of such fundamental topics.
To master this topic and more, students should join Skorminda — our classes are designed to build strong mathematical foundations.
—
Solving Linear Equations
Solving linear equations involves isolating the variable using inverse operations such as addition, subtraction, multiplication, and division.
Easy Question 1
Solve: $$ x + 5 = 12 $$
Solution:
Subtract 5 from both sides:
$$ x = 12 – 5 = 7 $$
Easy Question 2
Solve: $$ 3x = 9 $$
Solution:
Divide both sides by 3:
$$ x = \frac{9}{3} = 3 $$
Medium Question 1
Solve: $$ 2x – 4 = 10 $$
Solution:
Add 4:
$$ 2x = 14 $$
Divide by 2:
$$ x = 7 $$
Medium Question 2
Solve: $$ 5(x + 2) = 20 $$
Solution:
Distribute:
$$ 5x + 10 = 20 $$
Subtract 10:
$$ 5x = 10 $$
Divide by 5:
$$ x = 2 $$
Hard Question 1
Solve: $$ \frac{2x – 3}{4} = 5 $$
Solution:
Multiply both sides by 4:
$$ 2x – 3 = 20 $$
Add 3:
$$ 2x = 23 $$
Divide by 2:
$$ x = \frac{23}{2} $$
Hard Question 2
Solve: $$ 3(x – 2) – 4(x + 1) = 5 $$
Solution:
Distribute:
$$ 3x – 6 – 4x – 4 = 5 $$
Combine like terms:
$$ -x – 10 = 5 $$
Add 10:
$$ -x = 15 $$
Multiply by -1:
$$ x = -15 $$
—
Graphical Representation of Linear Equations
A linear equation in one variable can be represented on a number line to show the solution point accurately.
Easy Question 1
Plot the solution of \( x = -2 \) on a number line.
Solution:
Mark a point at -2 on the number line; that’s the solution.
Easy Question 2
Solve and plot: \( x + 1 = 0 \)
Solution:
\( x = -1 \). Plot point at -1.
Medium Question 1
Graph: \( 2x = 6 \)
Solution:
\( x = 3 \). Mark a single point at 3.
Medium Question 2
Graph: \( x – 7 = 2 \)
Solution:
\( x = 9 \). Mark point at 9.
Hard Question 1
Graph: \( \frac{x}{3} – 2 = -1 \)
Solution:
Add 2 → \( \frac{x}{3} = 1 \);
Multiply by 3 → \( x = 3 \). Mark point at 3.
Hard Question 2
Graph: \( 6 – 2x = 0 \)
Solution:
\( -2x = -6 \);
\( x = 3 \). Mark at 3 on number line.
—
Applications of Linear Equations
Linear equations are used to solve problems in finance, physics, engineering, and everyday situations like budgeting or travel planning.
Easy Question 1
If John has $x and spends $5, he is left with $10, find \( x \).
Solution:
\( x – 5 = 10 \Rightarrow x = 15 \)
Easy Question 2
A bag costs $x. Two bags cost $20. Find \( x \).
Solution:
\( 2x = 20 \Rightarrow x = 10 \)
Medium Question 1
A car rental costs $50 plus $x per hour. For 3 hours, cost is $110. Find \( x \).
Solution:
\( 50 + 3x = 110 \Rightarrow x = 20 \)
Medium Question 2
A phone plan charges $25 plus $5 per GB. If monthly bill is $50, how many GB used?
Solution:
\( 25 + 5x = 50 \Rightarrow x = 5 \)
Hard Question 1
An investor tripled his investment amount and lost $100, resulting in $200. Find original amount \( x \).
Solution:
\( 3x – 100 = 200 \Rightarrow x = 100 \)
Hard Question 2
A box contains \( x \) kg of sugar. After removing \( \frac{1}{2}x \), 3kg remains. Find total.
Solution:
\( \frac{1}{2}x = 3 \Rightarrow x = 6 \)
—
Importance of Studying Linear Equations in One Unknown
Understanding linear equations in one unknown is foundational for algebra, geometry, and calculus. They are the building blocks for higher mathematics and problem-solving in science and real-world applications.
Renowned mathematicians like Terence Tao and Maryna Viazovska study algebraic and linear structures in modern math research, showing the ongoing importance of such topics even today.
By mastering this topic, students gain:
– Logical problem-solving skills
– A base for higher education in mathematics
– Confidence in handling real-life mathematical problems
Ready to boost your skills? Join Skorminda classes to benefit from expert-guided lessons and interactive materials tailored for learners!
—
📚 Ready to Learn More?
Join Skorminda and unlock your full mathematical potential with our expert-designed courses and real-time problem-solving practice. Start your journey today!
No Comments