Chapter 1 Solving Linear Equations Answers

Solving linear equations is a fundamental skill in algebra, forming the basis for more advanced mathematical concepts. Understanding how to solve linear equations accurately and efficiently is crucial for success in mathematics and related fields. This comprehensive guide provides detailed answers and explanations for tackling linear equations, covering various techniques and strategies.

Understanding Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in the form ax + b = 0, where x is the variable, and a and b are constants. The goal is to find the value of x that satisfies the equation, making the left side equal to the right side.

Key Concepts

• Variable: A symbol (usually a letter like x, y, z) representing an unknown value.
• Constant: A fixed value that does not change.
• Coefficient: The number multiplied by a variable.
• Term: A single number or variable, or numbers and variables multiplied together.
• Equation: A statement that two expressions are equal.
• Solution: The value(s) of the variable that make the equation true.

Basic Techniques for Solving Linear Equations

1. Isolating the Variable

The primary goal in solving a linear equation is to isolate the variable on one side of the equation. This is achieved by performing the same operations on both sides to maintain equality.

Steps:

1. Simplify both sides: Combine like terms on each side of the equation.
2. Add or subtract: Use addition or subtraction to move constants to one side and variables to the other.
3. Multiply or divide: Use multiplication or division to isolate the variable.

Example 1:

Solve: 3x + 5 = 14

1. Subtract 5 from both sides:

   3x + 5 - 5 = 14 - 5

   3x = 9

2. Divide both sides by 3:

   3x / 3 = 9 / 3

   x = 3

Example 2:

Solve: 2x - 7 = -3

1. Add 7 to both sides:

   2x - 7 + 7 = -3 + 7

   2x = 4

2. Divide both sides by 2:

   2x / 2 = 4 / 2

   x = 2

2. Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable and exponent. This simplifies the equation, making it easier to solve.

Example 1:

Solve: 4x + 2x - 3 = 9

1. Combine like terms:

   6x - 3 = 9

2. Add 3 to both sides:

   6x - 3 + 3 = 9 + 3

   6x = 12

3. Divide both sides by 6:

   6x / 6 = 12 / 6

   x = 2

Example 2:

Solve: 5y - 3y + 8 = 16

1. Combine like terms:

   2y + 8 = 16

2. Subtract 8 from both sides:

   2y + 8 - 8 = 16 - 8

   2y = 8

3. Divide both sides by 2:

   2y / 2 = 8 / 2

   y = 4

3. Using the Distributive Property

The distributive property states that a(b + c) = ab + ac. This property is used to eliminate parentheses in an equation.

Example 1:

Solve: 2(x + 3) = 10

1. Distribute the 2:

   2x + 6 = 10

2. Subtract 6 from both sides:

   2x + 6 - 6 = 10 - 6

   2x = 4

3. Divide both sides by 2:

   2x / 2 = 4 / 2

   x = 2

Example 2:

Solve: -3(y - 4) = 9

1. Distribute the -3:

   -3y + 12 = 9

2. Subtract 12 from both sides:

   -3y + 12 - 12 = 9 - 12

   -3y = -3

3. Divide both sides by -3:

   -3y / -3 = -3 / -3

   y = 1

4. Equations with Fractions

Solving equations with fractions requires eliminating the fractions to simplify the equation.

Steps:

1. Find the least common denominator (LCD): Determine the smallest multiple that all denominators divide into evenly.
2. Multiply all terms by the LCD: This will clear the fractions.
3. Solve the resulting equation: Simplify and solve for the variable.

Example 1:

Solve: (x / 2) + (x / 3) = 5

1. Find the LCD: The LCD of 2 and 3 is 6.

2. Multiply all terms by 6:

   6(x / 2) + 6(x / 3) = 6(5)

   3x + 2x = 30

3. Combine like terms:

   5x = 30

4. Divide both sides by 5:

   5x / 5 = 30 / 5

   x = 6

Example 2:

Solve: (2y / 5) - (y / 4) = 3

1. Find the LCD: The LCD of 5 and 4 is 20.

2. Multiply all terms by 20:

   20(2y / 5) - 20(y / 4) = 20(3)

   8y - 5y = 60

3. Combine like terms:

   3y = 60

4. Divide both sides by 3:

   3y / 3 = 60 / 3

   y = 20

5. Equations with Decimals

Solving equations with decimals can be simplified by eliminating the decimals.

Steps:

1. Determine the highest number of decimal places: Identify the term with the most decimal places.
2. Multiply all terms by a power of 10: Multiply by 10, 100, 1000, etc., to eliminate the decimals.
3. Solve the resulting equation: Simplify and solve for the variable.

Example 1:

Solve: 0.2x + 0.5 = 1.3

1. The highest number of decimal places is 1.

2. Multiply all terms by 10:

   10(0.2x) + 10(0.5) = 10(1.3)

   2x + 5 = 13

3. Subtract 5 from both sides:

   2x + 5 - 5 = 13 - 5

   2x = 8

4. Divide both sides by 2:

   2x / 2 = 8 / 2

   x = 4

Example 2:

Solve: 0.05y - 0.15 = 0.2

1. The highest number of decimal places is 2.

2. Multiply all terms by 100:

   100(0.05y) - 100(0.15) = 100(0.2)

   5y - 15 = 20

3. Add 15 to both sides:

   5y - 15 + 15 = 20 + 15

   5y = 35

4. Divide both sides by 5:

   5y / 5 = 35 / 5

   y = 7

Advanced Techniques for Solving Linear Equations

1. Equations with Multiple Variables

Some linear equations may contain multiple variables. In such cases, the goal is to solve for one variable in terms of the others.

Example:

Solve for y: 3x + 2y = 12

1. Subtract 3x from both sides:

   2y = 12 - 3x

2. Divide both sides by 2:

   y = (12 - 3x) / 2

   y = 6 - (3 / 2)x

2. Equations with Absolute Value

Absolute value equations involve expressions within absolute value bars, which represent the distance from zero.

Steps:

1. Isolate the absolute value expression: Get the absolute value expression by itself on one side of the equation.
2. Set up two equations: One where the expression inside the absolute value is equal to the positive value, and one where it is equal to the negative value.
3. Solve both equations: Find the solutions for both equations.

Example:

Solve: |x - 3| = 5

1. The absolute value expression is already isolated.

2. Set up two equations:

   x - 3 = 5

   x - 3 = -5

3. Solve both equations:

   x - 3 = 5

   x = 5 + 3

   x = 8

   x - 3 = -5

   x = -5 + 3

   x = -2

So, the solutions are x = 8 and x = -2.

3. Equations with No Solution or Infinite Solutions

Some linear equations may have no solution or infinite solutions.

• No Solution: The equation leads to a contradiction, such as 0 = 5.
• Infinite Solutions: The equation simplifies to an identity, such as 0 = 0.

Example 1: No Solution

Solve: 2x + 3 = 2x - 1

1. Subtract 2x from both sides:

   3 = -1

This is a contradiction, so there is no solution.

Example 2: Infinite Solutions

Solve: 3(x + 2) = 3x + 6

1. Distribute the 3:

   3x + 6 = 3x + 6

2. Subtract 3x from both sides:

   6 = 6

This is an identity, so there are infinite solutions.

Practical Applications of Solving Linear Equations

Solving linear equations is not just an academic exercise; it has numerous practical applications in various fields.

1. Real-World Problems

Linear equations are used to solve real-world problems involving quantities, rates, and relationships.

Example:

A taxi charges a flat fee of $3 plus $2 per mile. If a ride costs $15, how many miles was the ride?

Let m be the number of miles. The equation is:

3 + 2m = 15

1. Subtract 3 from both sides:

   2m = 12

2. Divide both sides by 2:

   m = 6

The ride was 6 miles.

2. Physics

In physics, linear equations are used to describe motion, forces, and other physical phenomena.

Example:

The equation for the velocity v of an object with constant acceleration a after time t is v = u + at, where u is the initial velocity. If an object starts with an initial velocity of 5 m/s and accelerates at 2 m/s² for 10 seconds, what is its final velocity?

v = 5 + 2(10)

v = 5 + 20

v = 25 m/s

3. Economics

Linear equations are used in economics to model supply, demand, and cost functions.

Example:

The cost C of producing x units of a product is given by C = 5x + 100, where $100 is the fixed cost. If the selling price is $10 per unit, how many units must be sold to break even?

To break even, the cost must equal the revenue. Revenue R is given by R = 10x. So,

5x + 100 = 10x

1. Subtract 5x from both sides:

   100 = 5x

2. Divide both sides by 5:

   x = 20

20 units must be sold to break even.

Common Mistakes and How to Avoid Them

Solving linear equations accurately requires careful attention to detail. Here are some common mistakes and how to avoid them:

1. Incorrectly Distributing: Make sure to distribute to all terms inside the parentheses.

   Incorrect: 2(x + 3) = 2x + 3

   Correct: 2(x + 3) = 2x + 6

2. Combining Unlike Terms: Only combine terms that have the same variable and exponent.

   Incorrect: 3x + 2y = 5xy

   Correct: 3x + 2y = 3x + 2y

3. Forgetting to Multiply or Divide All Terms: When multiplying or dividing by a constant, ensure all terms are affected.

   Incorrect: (x / 2) + 3 = 5 => x + 3 = 10

   Correct: (x / 2) + 3 = 5 => x + 6 = 10

4. Sign Errors: Pay close attention to signs when adding, subtracting, multiplying, or dividing.

   Incorrect: -2x - 3 = 5 => -2x = 2

   Correct: -2x - 3 = 5 => -2x = 8

5. Not Checking Solutions: Always substitute your solution back into the original equation to verify its correctness.

Practice Problems

To reinforce your understanding, here are some practice problems with detailed solutions:

1. Solve: 4x - 7 = 9

   Solution: x = 4

2. Solve: 3(y + 2) = 15

   Solution: y = 3

3. Solve: (x / 3) - (x / 5) = 2

   Solution: x = 15

4. Solve: 0.5x + 1.2 = 2.7

   Solution: x = 3

5. Solve: |2x - 1| = 7

   Solution: x = 4, x = -3

Conclusion

Mastering the art of solving linear equations is a cornerstone of mathematical proficiency. By understanding the fundamental techniques, addressing equations with fractions and decimals, and avoiding common mistakes, you can confidently tackle a wide range of linear equation problems. Consistent practice and a thorough understanding of the underlying concepts will ensure your success in this essential area of mathematics.