Chapter 2: Linear Equations in One Variable
Overview of the Chapter
This chapter introduces students to the fundamental concepts of linear equations in one variable. It covers the methods to solve various types of linear equations and applies these methods to solve real-life problems. Understanding linear equations is essential as they form the basis for more advanced topics in algebra.
Introduction to Linear Equations
A linear equation in one variable is an equation that can be written in the form ax + b = 0, where x is the variable and a, b are constants. The solution to a linear equation is the value of x that satisfies the equation.
Summary of the Chapter
Detailed Summary
Basic Concepts
Linear equations are the cornerstone of algebra. They involve finding the value of a single variable that makes the equation true. Key operations used in solving linear equations include addition, subtraction, multiplication, and division.
Solving Linear Equations
To solve a linear equation, the goal is to isolate the variable on one side of the equation. This is achieved by performing inverse operations. For example, to solve 2x + 3 = 7, subtract 3 from both sides and then divide by 2:
Solution:
2x + 3 = 7
Subtract 3 from both sides:
2x = 4
Divide both sides by 2:
x = 2
Solving Equations Having the Variable on Both Sides
When the variable appears on both sides of the equation, the strategy is to collect all variable terms on one side and constant terms on the other. For example, to solve 3x – 5 = 2x + 4:
Solution:
3x – 5 = 2x + 4
Subtract 2x from both sides:
x – 5 = 4
Add 5 to both sides:
x = 9
Reducing Equations to Simpler Form
Simplifying equations before solving them can make the process easier. This involves combining like terms and simplifying fractions. The following table outlines different methods to reduce equations to their simpler forms:
| Method | Description | Example |
|---|---|---|
| Combining Like Terms | Combine terms that have the same variable raised to the same power. | 2x + 3x = 5x |
| Using Distributive Property | Expand expressions by multiplying the term outside the parentheses with each term inside. | 2(x + 3) = 2x + 6 |
| Clearing Fractions | Multiply both sides of the equation by the least common denominator to eliminate fractions. |
Original Equation: ½x + 3 = 7 After Multiplying by 2: x + 6 = 14 |
| Moving Terms Across the Equals Sign | Transfer terms from one side of the equation to the other by changing their signs. | x – 5 = 10 → x = 15 |
| Factoring | Factor out the greatest common factor from terms to simplify the equation. | 4x + 8 = 4(x + 2) |
Applications of Linear Equations
Linear equations are widely used in various real-life situations such as calculating distances, budgeting expenses, determining ages, and solving problems related to geometry and measurement.
Steps to Solve Linear Equations
Step-by-Step Procedure
Solving linear equations involves the following steps:
- Step 1: Simplify both sides of the equation if necessary.
- Step 2: Use inverse operations to isolate the variable.
- Step 3: Combine like terms.
- Step 4: Solve for the variable.
- Step 5: Check the solution by substituting it back into the original equation.
Common Mistakes to Avoid
Mistake 1: Ignoring the Sign Change
When moving a term from one side of the equation to the other, it’s crucial to change its sign. For example, if you move -3x from the right side to the left, it becomes +3x.
Mistake 2: Incorrect Fraction Handling
When dealing with fractions, ensure that you correctly find a common denominator when adding or subtracting them. For instance, adding ½ and ⅓ requires a common denominator of 6:
Calculation:
½ + ⅓ = ⅗ + ⅖ = ⅚
Real-Life Applications
Age Problems
Age-related problems often require setting up linear equations based on the relationships between the ages of individuals at different times. For example:
Problem: In 5 years, John’s age will be twice his age 3 years ago. Find John’s current age.
Solution:
Let John’s current age be x.
In 5 years: x + 5
3 years ago: x – 3
Equation: x + 5 = 2(x – 3)
Solve:
x + 5 = 2x – 6
5 + 6 = 2x – x
x = 11
Distance Problems
Distance problems utilize the formula Distance = Speed × Time to set up linear equations. For example:
Problem: A car travels at a constant speed. If it travels 150 km in 3 hours, what is its speed?
Solution:
Let the speed be x km/h.
Distance = Speed × Time → 150 = x × 3
Solve: x = 150 ÷ 3 = 50 km/h
Important Examples and Cases
Example 1: Solving a Basic Linear Equation
Problem: Solve for x: 2x + 3 = 7
Solution:
Subtract 3 from both sides:
2x = 4
Divide both sides by 2:
x = 2
Example 2: Word Problem Involving Linear Equations
Problem: Maria has twice as many apples as Tom. Together, they have 18 apples. How many apples does Maria have?
Solution:
Let the number of apples Tom has be x.
Maria has 2x apples.
Equation: x + 2x = 18
Solve: 3x = 18 → x = 6
Maria has 2x = 12 apples.
Key Terms and Concepts
- Linear Equation: An equation of the first degree, meaning it contains no exponents higher than one.
- Variable: A symbol, usually x or y, that represents an unknown quantity.
- Constant: A fixed value that does not change.
- Solution: The value of the variable that satisfies the equation.
- Inverse Operation: An operation that reverses another operation (e.g., addition and subtraction).
- Like Terms: Terms that have the same variable raised to the same power.
Additional Value Addition
Tips for Solving Linear Equations
- Always perform the same operation on both sides of the equation to maintain equality.
- Keep the variable on one side and constants on the other for easier manipulation.
- Check your solution by substituting it back into the original equation.
- Be cautious with negative signs when moving terms across the equals sign.
- Practice with different types of equations to build confidence and proficiency.
Chronology of Key Concepts
Example 1: Solving a Fractional Linear Equation
Problem: Solve for x: ½x + ⅓ = ⅚
Solution:
Find the least common denominator (LCD) for the fractions, which is 6.
Multiply both sides by 6 to eliminate the fractions:
6 × ½x + 6 × ⅓ = 6 × ⅚
Simplify:
3x + 2 = 5
Subtract 2 from both sides:
3x = 3
Divide both sides by 3:
x = 1
Example 2: Word Problem Involving Geometry
Problem: The perimeter of a rectangle is 50 cm. The length is 5 cm more than twice the breadth. Find the dimensions of the rectangle.
Solution:
Let the breadth be x cm.
Length = 2x + 5 cm.
Perimeter of a rectangle = 2(length + breadth).
Equation: 2(2x + 5 + x) = 50
Simplify:
2(3x + 5) = 50 → 6x + 10 = 50
Subtract 10 from both sides:
6x = 40
Divide both sides by 6:
x = 20/3 cm
Length = 2 × 20/3 + 5 = 40/3 + 15/3 = 55/3 cm
FAQs on Chapter 2: Linear Equations in One Variable
A linear equation in one variable is an equation that can be written in the form ax + b = 0, where x is the variable and a, b are constants. The solution to the equation is the value of x that makes the equation true.
To solve a linear equation in one variable, isolate the variable on one side of the equation using inverse operations. This typically involves adding, subtracting, multiplying, or dividing both sides of the equation.
Like terms are terms that have the same variable raised to the same power. In a linear equation, like terms can be combined to simplify the equation.
Isolating the variable means rearranging the equation so that the variable is on one side of the equation and the constants are on the other side. This is done to solve for the variable.
To clear fractions in a linear equation, multiply both sides of the equation by the least common denominator (LCD) of the fractions. This eliminates the fractions and simplifies the equation.
The distributive property states that a(b + c) = ab + ac. It is used in solving linear equations to eliminate parentheses by distributing the multiplication over the addition or subtraction inside the parentheses.
Yes, linear equations can have variables on both sides. To solve such equations, move all terms with the variable to one side and the constant terms to the other side.
The solution to a linear equation is the value of the variable that makes the equation true when substituted back into the original equation.
Adding or subtracting the same value from both sides of an equation maintains the equality of the equation. This is often used to move terms from one side of the equation to the other.
Multiplying or dividing both sides of an equation by the same non-zero value maintains the equality of the equation. This is used to simplify the equation and solve for the variable.
A constant is a fixed value that does not change. In a linear equation, constants are the terms without variables.
Common mistakes include ignoring the sign change when moving terms, incorrect fraction handling, and forgetting to distribute multiplication over addition or subtraction.
To solve an equation with a variable on both sides, first move all the terms containing the variable to one side and all constants to the other side. Then solve for the variable.
The first step in solving a linear equation is to simplify both sides of the equation, which may involve combining like terms or distributing multiplication.
Yes, a linear equation can have no solution if, after simplifying, you get a false statement, such as 0 = 5. This means that there is no value of the variable that will satisfy the equation.
Yes, a linear equation can have infinitely many solutions if, after simplifying, you get a true statement that holds for any value of the variable, such as 0 = 0.
Checking your solution ensures that the value you found for the variable actually satisfies the original equation. This helps to catch any errors made during the solving process.
To solve linear equations with fractions, first find the least common denominator (LCD) and multiply every term by it to eliminate the fractions. Then solve the resulting equation.
The coefficient is the number multiplied by the variable in a linear equation. It determines the rate at which the variable affects the equation. For example, in 3x = 9, 3 is the coefficient of x.
Negative coefficients can be handled by performing inverse operations, such as dividing both sides of the equation by the negative coefficient to isolate the variable.
Linear equations are used in various real-life applications such as calculating distances, budgeting, determining ages, and solving problems related to geometry and measurement.
In age problems, linear equations are used to express the relationship between the current, past, or future ages of individuals. The equation is then solved to find the unknown age.
Distance problems involve finding the distance, speed, or time when two of these variables are known. Linear equations are used to solve for the unknown variable using the formula Distance = Speed × Time.
To solve word problems involving linear equations, translate the problem into an equation by identifying the variable and writing the equation based on the relationships described. Then solve for the variable.
Some tips include: Always perform the same operation on both sides, simplify the equation first, check your solution, be cautious with signs, and practice with different types of problems.
MCQs on Chapter 2: Linear Equations in One Variable
1. What is a linear equation in one variable?
2. Which of the following is a linear equation?
3. What is the solution to the equation 2x – 3 = 7?
4. If 3x + 4 = 10, what is the value of x?
5. What is the result of solving the equation 5x – 15 = 0?
6. Which of the following represents a linear equation in one variable?
7. What is the value of x if 4x – 7 = 9?
8. Solve: x/2 + 3 = 7
9. If 6x + 3 = 21, what is x?
10. Which equation has no solution?
11. What is the solution to 3x + 9 = 0?
12. What is the coefficient of x in the equation 5x – 4 = 16?
13. Solve: 4x – 16 = 0
14. Which of the following is the standard form of a linear equation in one variable?
15. What is the value of x if 2x + 3 = x – 4?
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