Solving Linear Equations in One Unknown

Introduction

Key Concepts

Definition of Linear Equations

A linear equation in one unknown is an algebraic statement that represents a straight line when graphed on a coordinate plane. It has the general form:

where \(x\) is the unknown variable, and \(a\) and \(b\) are constants with \(a \neq 0\). The solution to this equation is the value of \(x\) that makes the equation true.

Solving Basic Linear Equations

To solve a basic linear equation, the goal is to isolate the variable \(x\). This involves performing inverse operations to eliminate constants and coefficients attached to \(x\).

Example 1:

Solve for \(x\):

Solution:

• Subtract 5 from both sides:

• Divide both sides by 2:

Therefore, the solution is \(x = 4\).

Equations with Fractions

Linear equations may sometimes involve fractions. In such cases, it's essential to eliminate denominators to simplify the equation.

Example 2:

Solve for \(x\):

Solution:

• Add 2 to both sides:

• Multiply both sides by 4 to eliminate the denominator:

• Divide both sides by 3:

Therefore, the solution is \(x = 4\).

Equations with Variables on Both Sides

Sometimes, variables appear on both sides of the equation. The technique involves bringing all \(x\) terms to one side and constants to the other.

Example 3:

Solve for \(x\):

Solution:

• Subtract \(2x\) from both sides:

• Add 3 to both sides:

• Divide both sides by 3:

Therefore, the solution is \(x = 4\).

Understanding the Solution Set

Linear equations in one unknown have either one solution, no solution, or infinitely many solutions.

• One Solution: Occurs when there is a unique value of \(x\) that satisfies the equation.
• No Solution: Happens when the equation simplifies to a false statement, such as \(0 = 5\).
• Infinitely Many Solutions: Arises when the equation simplifies to an identity, like \(0 = 0\), indicating that any value of \(x\) satisfies the equation.

Example 4:

Solve for \(x\):

Solution:

This simplifies to a true statement regardless of \(x\), hence infinitely many solutions.

Application of Linear Equations

Linear equations are widely used in various fields such as physics, economics, and engineering. They help model relationships where one variable depends linearly on another.

Example: Calculating the cost \(C\) of buying \(n\) notebooks, if each notebook costs £2 and there is a fixed cost of £5.

To find out how many notebooks you can buy with £17:

Thus, you can buy 6 notebooks.

Graphical Interpretation

A linear equation in one variable can be represented graphically on a number line. The solution is the point where the equation balances.

Example 5:

Solve for \(x\):

Solution:

Graphically, the point \(x = 10\) is the solution.

Checking Solutions

Always verify solutions by substituting them back into the original equation to ensure they are correct.

Example 6:

Equation:

Solution:

Check:

The solution is verified.

Special Cases

Understanding special cases helps in identifying unique solutions or the lack thereof.

• No Variable: An equation like \(5 = 5\) has infinitely many solutions.
• Zero Coefficient: If the coefficient of \(x\) is zero, the equation may have no solution or infinitely many solutions.

Example 7:

Solve for \(x\):

Solution:

Here, \(x\) can be any real number.

Word Problems Involving Linear Equations

Word problems translate real-life situations into mathematical equations. Let's consider an example.

Example 8:

A car rental company charges a fixed fee of £50 plus £20 per day. How many days can you rent the car for £170?

Solution:

Let \(d\) be the number of days.

You can rent the car for 6 days.

Advanced Concepts

Theoretical Foundations and Proofs

Delving deeper into linear equations involves exploring their theoretical underpinnings, such as the principles of equality and algebraic manipulation. Understanding why the methods used to solve these equations work is crucial for higher-level mathematics.

The fundamental principle is that of maintaining equality while performing inverse operations:

• Addition: If \(a = b\), then \(a + c = b + c\)
• Subtraction: If \(a = b\), then \(a - c = b - c\)
• Multiplication: If \(a = b\), then \(ca = cb\) (provided \(c \neq 0\))
• Division: If \(a = b\), then \(\frac{a}{c} = \frac{b}{c}\) (provided \(c \neq 0\))

Theorem: A linear equation in one variable has exactly one solution unless it is an identity or a contradiction.

Proof:

Consider the general form of a linear equation:

Solving for \(x\):

Since \(a \neq 0\), there exists exactly one solution for \(x\).

If \(a = 0\), then the equation becomes \(b = 0\). If \(b = 0\), it's an identity with infinitely many solutions; else, it's a contradiction with no solution.

Systems of Linear Equations

While focusing on one variable, understanding systems involving multiple variables can deepen comprehension of equations' relationships and dependencies.

However, for the purpose of this section, we focus on extensions of single-variable equations.

Parametric Linear Equations

Introducing parameters allows exploration of families of equations and their solutions.

Example:

Consider the equation:

For different values of \(k\), the solution changes:

This showcases how parameters influence the solution set.

Graphing Equations and Slope-Intercept Form

While graphing linear equations in one variable is trivial (as they are constants on the number line), exploring slope-intercept form helps in transitioning to two-variable linear equations.

The standard slope-intercept form is:

Where \(m\) is the slope and \(c\) is the y-intercept. Understanding this form is essential for graphing and analyzing linear relationships in higher dimensions.

Applications in Optimization Problems

Linear equations play a vital role in optimization, where determining the optimal value involves solving linear equations and inequalities.

Example:

A company needs to produce at least 100 units of a product. The cost function is \(C(x) = 5x + 200\), and the revenue function is \(R(x) = 8x\). Find the break-even point.

Solution:

However, since production needs to be at least 100 units:

At \(x = 100\):

Therefore, the company makes a profit of £100.

Linear Equations in Vector Spaces

In more advanced studies, linear equations extend into vector spaces, forming systems that represent planes or higher-dimensional analogues. Though beyond the scope of Cambridge IGCSE, gaining an introductory understanding can be beneficial for further mathematical pursuits.

Interdisciplinary Connections

Linear equations intersect with various disciplines, enhancing their applicability and relevance.

• Physics: Representing motion with constant speed.
• Economics: Modeling cost functions and supply-demand relationships.
• Engineering: Designing systems with linear constraints.
• Computer Science: Algorithms relying on linear computations.

Example:

In physics, the equation \(d = vt + s\), where \(d\) is distance, \(v\) is velocity, \(t\) is time, and \(s\) is initial position, is a linear equation in terms of \(t\).

Complex Problem-Solving Techniques

Solving complex linear equations requires strategic approaches, such as substitution, elimination, and graphical methods.

Multi-Step Equations:

Equations that require several steps to isolate the variable.

Example:

Solve for \(x\):

Solution:

• Expand the equation:

• Add 8 to both sides:

• Divide by 6:

Equations Involving Absolute Values

Linear equations with absolute values consider both positive and negative scenarios of the variable.

Example:

Solve for \(x\):

Solution:

This splits into two equations:

• \(2x - 3 = 7\)
• \(2x - 3 = -7\)

Solving the first equation:

Solving the second equation:

Therefore, \(x = 5\) or \(x = -2\).

Equations with Nested Variables

Equations might include variables within exponents or other expressions, requiring advanced techniques to solve.

Example:

Solve for \(x\):

Solution:

Therefore, \(x = 4\).

Use of Technology in Solving Linear Equations

Graphing calculators and computer algebra systems (CAS) can assist in solving and visualizing linear equations, enhancing understanding and efficiency.

Example:

Using a graphing calculator to solve \(2x + 5 = 13\) involves plotting the function \(f(x) = 2x + 5\) and identifying the point where \(f(x) = 13\). The x-coordinate at this intersection is the solution.

Comparison Table

Summary and Key Takeaways